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ATREATISE ON DIFFERENTIAL EQUATIONS BY ANDHEW ETJSSEIL FORSYTE, MA, F.E.S., FEILOW .AND AqSISTAlH? TUTOK OPTBINITS COLLEGE, OAMBBIDGE. SECOND EDITION. MACMILLAN A.ND 00. * 'ANDttEW TOEK. 1888 .f\fTrnnstlsitian. PEERAGE. INthepresentrelume Ihave tried tomake the discussion ofthevariouspartsofthesubject,which are heregiven,asfullaspossible; andthere willbefound much which hashitherto notappeared exceptinmathe- maticaljournals. Atthesame time, thetreatise does notprofesstohecomplete. Amongthepartsomitted aretheinvestigations "by.Fuchs'-on/the integrationof linear differential'equations,thoseof;Konigsbergeron theirreducibilityofdifferentialequations,thediscussion ofPfaffsequation-'therecent researches ofHermite and Halphen, andthegeometrical3 applicationsofthehyper- geometricseries byKlein; onlyavery slightsketch ofJacobi's method forpartialdifferentialequationsis attempted,andthere isnoindication ofthemethods of Cauchy,LieandMayer. These, andothers hereomitted, Ihopetogiveinanother volume atsome future date. Whilewritingthisvolume Ihave consulted many authorities intheshapeoftreatises, memoirs andtext- books;and,thoughitisimpossibleto'giveindetail every reference, Iwish inparticulartomention, as havingfeeen ofgreat use, Boole's Treatise and his Supplement, Moigno, ImschenetskyandMansion; and vi PREFACE. Ihave used, toaslighterextent than these, Gregory's Examples,Serret andDeMorgan. Manyreferences to originalmemoirs willbefound invariouschapters. There occur, scatteredthroughoutthebook,many examples, amountinginnumber tomore thaneight hundred. Most ofthese aretaken fromUniversity and CollegeExaminationpaperssetinCambridgeatvarious times;some arenew,andmanyofthem areresults extracted frommemoirs which havebeen consulted. In thecase ofthe last, theoriginal authority is,Ithink, alwaysindicated. Icannothope that,amongsomany, allresultsgivenarecorrect and allequationssetare soluble;and Ishall begladtoreceive corrections ofany mistakesactuallyfound. Inconclusion, Iwish toexpressthevery great obligationsunder which Ilietomyfriend andformer tutorMrH.M.Taylor,ofTrinity College, Cambridge, forhiskindness inthe revision oftheproof-sheets. Hehascaused theremoval ofmanyobscurities and hasmademanyvaluablesuggestionsofwhich Ihave continuallyavailedmyself. Mythanks are alsodue tomyfriendMrJ.M.Dodds, ofStPeter'sCollege, Cambridge,forhiskindness inreadingsome ofthe earlysheets. A.RFORSYTE. TRINITY COLLEGE, CAMBRIDGE, Re-n+om'hov >REFACE TOTHESECOND EDITION. THIS edition willbefound todiffervery slightlyfrom first. InitspreparationIhave beenmuchhelped bhekindness ofmanyfriends andcorrespondents who esentmenotification ofmistakes andmisprints. Mythanks arespeciallyduetoDrHermann Maser Berlin forthehonour hehasdonemeintranslating book intoGerman. A.KF. TRINITY COLLEGE, CAMBRIDGE, CONTENTS. CHAPTER I. INTRODUCTION. AET. PAGE 14. Formation ofDifferential Equations andcharacter ofsolutions . 1 5. What istobeconsidered asolution 5 6. Definitions 7 7.8. Number offirstintegrals ofagiven equation 8 9,10.Lemmas relating tofunctionality 11 CHAPTER II. DIFFERENTIA! EQUATIONS OFTHEFIRST ORDER. 1L General equation ofthefirstorder ..15 12, Anequationoffirstorderand firstdegree hasonlyoneprimitive.15 13. Standard I.:variables separable 16 14,15. Standard II. :linear equation 18 16,17. Standard EH. :homogeneous equation 20 Id. Standard IV. :onevariable absent 28 10. Standard V. :equation ofthe 71thdegree 25 2022.Standard VI. :Olairaut's form 27 23,24. Existence ofSingular Solutions 30 25. Derivation oftheSingular Solution from theprimitive. . .32 26,27.Envelope locus, nodal locus, cuspidal locus * 38 28. Derivation oftheSingular Solution from thedifferential equation ; introduction oftao-loous 34 29. Envelope locus istheonlyonewhose equationisasolution . .35 30. Anequation ofthenthdegree hasnotnecessarily aSingular Solution 35 Miscellaneous Examples 39 CONTENTS. CHAPTER III. GENERAL LINEAR EQUATION WITH CONSTANT COEFFICIENTS. ART. PAGE 3137. Theorems indifferentiation andintegration 43 38. Form ofthelinear equation 48 39. Itsprimitive consists oftwoparts 49 4042. Generalproperties 49 43 45.Derivation oftheComplementary Function 52 46. Derivation oftheParticular Integral insometypical forms . .57 47,48. Solution ofthehomogeneous linear equation 66 Miscellaneous Examples 69 CHAPTER IV. MISCELLANEOUS METHODS. 49. Limitation ofmethods inthechapter 72 50. Solution ofy^= function ofx 72 51. Solution ofy"=function ofy 73 52. Solution ofy= function ofy^-u 74 53. Solution of2/(n)=function of2/(n~3)75 54. Depressionoforderwhen onevariable isabsent.... 77 55. Equations possessing generalised homogeneity.... 79 56. Exact equations which arelinear 82 57. Exact equations which arenotlinear 84 58. General linear equation ofsecond order isintegrable when any single integral ofasimpler form isknown.... 86 59.60. Reduction ofequation tonormal form inwhich theonly algebraical coefficient isaninvariant 88 61,62. Equation ofthird order satisfied byquotient oftwosolutions;the Schwarzian derivative 90 63. Solution ofparticular cases ofthelinear equation bychange of independent variable 93 64. Conditions forequivalence oftwogiven equations.... 95 6567. Method ofVariation ofParameters appliedtoequationsofsecond order 98 68. Solution incase ofparticular form oftheinvariant . . .104 69. Integration byresolution ofthedifferential operator. ..106 70. Form ofequation usedbySirWilliam Thomson.... 108 7173. Condition thatanumber ofparticular integrals ofthegeneral linear equation should beindependentisthenon-evanescence ofacertain determinant 109 74. Value ofthisdete inant . Ill 75. Derivation oftheParticular Integral bythemethod ofVariation of Parameters 112 76. Depressionoftheorderwhen particular integrals areknown . .115 CONTENTS. XI ART. 77. Solution when allparticular integrals butoneareknown . .116 78. Geometrical application; trajectories......"110 79. General trajectory......... lift 80 82. Orthogonal trajectories........ ISO- Miscellaneous Examples....... 125 CHAPTER V. INTEGRATION INSERIES. 83,84. Possibility ofsolution byapproximationintheform ofacon- verging series......... 182 85. Solution of** +*a-ly=..... W5 86. Form ofsolution when azero factor enters intothedenominator ofacoefficient hitheseries....... 18J> 87,88. Case in-which there isasolution consisting ofafinitenumber of terms........... 141 89. LEGENDRE'S equation......... 143 90. Thesolution y=P n......... 144 91. Thesolution y=Qn......... 147 92. Different cases tobeconsidered....... 14H 9395. Primitive inthecaseswhen there ionlyaHin^lu particular solution obtained, that is,when 2//.iHanoddintogur. .14H 96. Differential relation between PnandQn..... 155 97-99. Modified form ofthisrelation....... l.W 100. BESSEL'S equation......... 151) 101,102. Thesolutions y=7"nandy=.7^....... 100 103. Proportion ofthefunctions J....... 103 104. Thesolution y=YQwhenniszero...... 104 105. Thesolution y=Ynwhennisintegral..... 105 106. Differential relation between Jnand r7_n..... 107 107. Deduction ofBessel's equation from Lagendrc'H equation. .108 108. BICCATI'B equation......... 170 109.110. Cases inwhich thisequation andamore goiioral form arein- tegrabhs infinite terms........ 170 111. Reduction oflliccati's equation toBassel's equation. . 17 112. Symbolical flotations......... 170 Miscellaneous Examples....... 17K CHAPTER VI. THEHYPERGEOMETRIC SELtlKS.* 113. Definition ofthe series; special cases...... 185 114.115. Differential equationofthesecond order satisfied bytheHerics; primitive oftheequation....... 180 116. Normal formofthedifferential equation..... 188 CONTENTS. 117, 118. Equations Hulmidiarytiithoduduction ofparticular solutions of rthoequation.......... 189 119. Sixvalues ofthevariable! olcmuiit . t^ 120,121. getof 24particularHolutioiw . ._... 191 122. Lemma relatingtoctnvtirin HorioH ..... 194 123. Division ofthe 2-1solutions into KixclaRHoH offour each . .195 124. TheclaHHCH botwceii thunqualmembers ofwhich alinear relation exists........... 196 125. ExpressionforthoKCTICK witlithevariable argument made unity 197 12G. GauflH'B 11function......... 198 127129. Determination ofthucoiwtantH inthelinear relations of124 .200 130. Tho Bchwar/.ian derivative forthodifferential equation tobe appliedtoobtain theCRHCH ofintegration inafinite form .204 181. CMC I.; (ft+l)B*-4ii........ 205 132. Oao II.;x(**-tofij:i-ljn-(**+aa,y"- l):l.... 209 133. CaHe III.;combination ofI.and II. ..... 211 134. BcfcTHiifiOH tooriginal inrmoirH . . .... 212 MiHCulluncotiH KxiunplcH-..... 213 CIIAPTKH. VIE. HOLI;TION UYDKFINITK INTKORALS. (JHAPTKIl Vllf. OEDINAUY KQUATION8 WITHUOKKTHANTWO VARIABLES. 14C. ETJLKK'H quntion ;Hiclirlot'M mothod ofintegration. .239 147. Caachy'H method ofintcKration...... .241 148, 149. Gcneraliwiticm ofKith-r'M nitiAiicm ;method c^fintegrationdueto CONTENTS. Xlll PAGE TOTAL DIFFERENTIAL EQUATIONS; formation from given primitive 249 Such equations donotnecessarily imply theexistence ofasingle" primitive.......... 2;>0 Relation between coefficients inPdx+Qdy+Rdz=Q thatasingle primitiveshould exist 250 Method ofintegration -when thisrelation issatisfied . . .251 Method ofintegration when thisrelation isnotsatisfied . .253 L56. Comparisonoftheprimitivesinthetwocases.... 255 Geometrical interpretation ;thelocus representedisafamily of curves 258 L59.Onanyarbitrary surface there isasinglyinfinite Boriew ofcurves 25B L61. Inaparticularcase allthecurveslying onsome onosurface are included inthelocus andtherefore thesurface itself inin- cluded 200 Identification ofthiscasewith that of153 . . ..200 Total equations innvariables; conditions tobeBatiwrlod that suchanequation should bederivable from asingle primitive 201 Method ofintegration when these conditions aresatisfied . .202 Case ofequations which arenotlinear 203 SIMULTANEOUS EQUATIONS ;cases inwhich they arise . . .205 Method ofintegrationoflinear equations with coiiHtant co- efficients 20G Relations between thearbitrary constants 207 Number ofindependent arbitrary constantH inthetfonunil case .20B Forms ofHolution(i)forimaginary rootfi, (ii)foroqiml roots .20rt Special forniB ofHolution 200 Simultaneous equations with variable cooiliciontH;HiiJttciont to consider equations offirstorder 272 When inthemodified form there aremdependent variables, tho solution canbemade todepend upon that ofunordinary equation ofthe ?/tUlorder 273 Case ofsimplification 275 Integration oftlioequations ofmotion ofaparticle moving under acentral force 27H Kxamplen 2H2 CHAPTER :iX. PARTIAL DIFPERKNTIAL EQUATIONS OFTJEK FIKHT ORDER. Notation anddefinitionw.... ^... 280 Claflmfication ofintegrals ofapartial differential equation. . 2fi7 TheComplete Integral......... 2H7 TheSingular Integral 28H TheGeneral Inte^al 200 Every solution ofthoequation isincluded innomo oneoftho xiv CONTENTS. ART. PAOE 182. Geometrical interpretation inthecase inwhich there aretwo independent variables 204 183. Derivation oftheSingular Integral from thedifferential equation 290 184. LAOIIANGE'S LINEAR EQUATION ;thedifferential equation equivalent to0(/t,v)=0 209 185. Derivation ofintegral ofPp+Qq=R 800 180. This integralisthemost general 301 187. Particular solutions oftheequation 303 188. Theform ofequations which haveanintegral (u,v)=Q . .303 180. Generalisation tothecaseofnindependent variables . . .303 190. STANDARD FORMS 306 191. Standard!.:\l/(p,q)=Q 300 192. Geometrical interpretation of^(p,q)= 308 193. Standard II. :x(*JP <7)= 308 194. Geometrical interpretation oftheintegral 310 195. Standard III. :</(*, j/>)=V'(?/, ff) 310 196. StandarclIV.: z=pt+qy+4>(p, q) 312 197. Duality ofpartialdifferential equations 313 198. This duality correspondstotheprinciple ofduality ingeometry.315 199. Determination, inspecial canes, ofthearbitrary function which occurs intheGeneral Integral 310 200. PrincipleofCHARPIT'H MKTHOI> fortheintegration ofthegeneral equation containing twoindependent variables . . .317 201. Deduction ofthesubsidiary equations used inthismethod . .318 202. Be-enunciation oftboresult of201 319 203. TheStandard forms areparticular cases inwhich Charpit's method isimmediately effective 321 204. Lagrango's linear equationisaparticular case . . ..3*22 205. Proof thatStandard I.isaparticular case IW2 206. Proof thatStandard II.isaparticular case 323 207. Proof thatStandard III. isaparticular case.... 324 208. TheGENERAL EQUATION ofthe firstorder withnindependent variables 325 209. Itcanalways boreplaced byonewhich does notcontain the dependentvariables 325 210. Principle ofthemethod usedbyJACOBI fortheintegration ofthe general equation 320 211. Deduction ofthenecessary subsidiary equations.... 327 212. These equations arcsufficient 329 213. Formulation oftherule towhich themethod leads . . .331 214. Lemma onfunctions connected withthesubsidiary equations.332 215 222. Integrationofthesubsidiary equations 334 223. List ofauthorities onpartialdifferential equations. . .342 Examples^fJacobi's method 343 224. SIMULTANEOUS PARTIAL EQUATIONS 847 225. Case inwhich thenumber ofequations given isequal tothenum- berofindependent variables 348 226. Case inwhich thenumber ofequations giveninlewsthan the number ofindependent variables 349 CONTENTS. XV OKAFIER X. PARTIAL DIFFERENTIAL EQUATIONS OKTIIKHKUOXD AND IIUillKK OKDKItS. ART. I'AUK 227. Notation and definitional .'Jl>( 228. Simple cnHGB oftheequation llr+N*+'J't~l/'.... :J57 229. MONGK'H PKOCBHH ofintegration of//+#*+77K . . .:."H 230.281. Investigation oftheform ofequation towhich thinprocuHH may boapplied35H 232. Deduction ofintermediary integralf/*rH-*'*I'/'* -I-f/(rf -H")KMO 233. When r/IHy.ero, twointermediary integrate arc, intfcnoral, obtained M2 234. When ITinnotzero, twointermediary integral are alHo,in general, obtained .'{ft-'J 235. Deduction of^cnoral integral fromanyintorrrwdiury intfigrril.JMJ4 236. When twointermediary inte^ralK arcdbtuinud, thoymay bo treated ansiniultuneouH (itjuationH in2>nndq. . . .Nift 237,238. Proof ofthupropoHitionof gy-'JO :r 239. Summary ofth<;itiotliod ofHolution .'JOH 240,241. ProccHScs tolu:adoptedinfailing caKes .'i70 242. Principleoffinality:J7 24.'!. IjAi'iiirK'H tnLMHforrniiLion oftinslinnar (iciuiilion ;orinIbrin . .'Ml 2-14. Two inti^rul'lt! C.IIHOK ofthetntimforiiKMl (ujuuiion. . . .'t?H 2-15. Further tniiisformution v,-}n:n thecondition*! of^'241arcnot HatiHfied ;t7i) 21(1. Alternative form ofthotranHfonnation JtHl 247. POXKKON'K method foraHpectal form oflhhomo^onooiiH o<ination .'JH'J 24H... LINKAII ICgnATinN WITHCUNHTANT COKKKICIKNTH.... JiH.'j 219. Thecomplementary function inthoduto inwhich dilTerenlial ciii>Hic.i(!iit.Konly*ftin-lbunlnr occur UH4 2.10. I'articuliir mtuum! inthineaw. :w/5 251. Method ofprocfiedingforthecomplcnusnUry function ofthoin(nit Kc:niral form ftKH 2.72. Modification ofthecomplementary function inHpeciul na^cH .MHO 253. Deduction oftheparticular integral JJU1 254. ClaHH ofhoriin^cn<;fiu.'t <ujuutu;nH :KI2 2">5. MiKuclluneoiiH mothodri :J!) I 2/iO. Holution oftliuonmLinn .-.*..,intwoftmtm . . .IMM1 rit tif* 257. PrtHif that th?H<i twoforniH areequivalent :M)0 25s. Kynthntic Holution intheform ofadcflnitft Integra). . .\VM 250. Holution inthinform byaKyznbolioal tnuthwl.... 3!)H 2<U). SOLUTION isHKHIK;thoequation. -i- .a-|-,a--0 . . .101 201204. Special formn ofnolutionH ofthiswjunt inn 101 JSflfi. A\fi.t-ni.-' \TfTifnt. f/itin.ffinntmti *ifK*>'W A(\l\ XVI CONTENTS. ABT. PAGE 266. Modification ofthesubsidiary equations 407 267.rEquations tobesatisfied byafunctionW 408 268. When thisfunctionWisknown, asolution oftheequationis given409 269. General form ofW ; .410 270.271. Generalisation ofanintegral containing anumber ofarbitrary constants bythemethod ofvariation ofparameters. .410 Miscellaneous Examples....... 417 CHAPTER i. INTRODUCTION. 1.WHEN onevariable quantity yisnfunction ofanother variablequantity 0;,tho relation between thotwomaybeexhibited bymeans ofanequationHiich us (*,y)=o. Inthisequut,inncoriHtan t.smayoccur;letoneof.such constants be denoted bya.Iftheequationbesolved foryinterms ofu\this constant awillenter into theexpressionfor\j;and,bytaking different values for</,there, will ingeneralbe.obtained anumber ofcorrespondingvalues fory.Ifitbedesired toindicate inthe fundamental relation thefactthat tin*value ofydependsonthat ofa,thismaybedonebywritingtheuhovuequationintheform ^fey. '0-0 (i). Now itisfM>Hsibl(jtoderive, from thisequation another, which shall include allthevulues of;/,which canboobtained byas- signingallthepossiblevalues totheconstant a.The? differentia,! Cfxifficient ofywithregardtoxIKgiven by r>d> <)d>dij ^ ',*j""0(u),cte3ydx'* inwhich^and -indicatepartialdifferon tuition with regardt*> xandyrespectively. Ecjuation (ii)willingeneralinvolve the ltt/'ftMnj tti/l\ fi/l iTl*rf Vi^i^.rt fViiiuci *tin iitfitin. 2 INTRODUCTION.[1. tions theconstant beeliminated, theresult oftheelimination will beof-theform where/isadefinite functiondependingontheform ofthe function<j>inequation (i).Nowequation (iii)isone,which includes allthevalues of?/,which canarise from(i);for,while itisderived from thetwoequations (i)and(ii),ineach of which aoccurs, yetoftheparticularvalue ofthisquantity nospecialaccount istaken and,wereanyother constant asa substituted forainallthestepsoftheelimination, theresult would bethesame, since theconstant ismade todisappearfrom theresult. Inthesameway,ifydependedontwoconstants a.andbin amanner defined byanequation *(*, y,,&)=o, and ifthoequationswhichgivetho firstandsecond differential coc.fticientH of?/withregardtoxwore written down, thetwocon- stants aand Icould beeliminated andtheresulting equation would beoftheform Inallcases thefunctions /andFcanbededuced(bymethods of theDifferential Calculus andofHigher Algebra) when theforms fj>and <E>aregiven. Inparticular,ifsuchaformbe (as,y)=a, fromwhich aistobeeliminated, then, astheequation embracing allthevaluesy,wehave atonce <+dv=odxdydx9 nofurther eliminationbeingneeded. Thus, forexample,theequation f leads totheequation 1.]INTRODUCTION. 3 which isthegeneral equationofallparabolas havingthesame axisandvertex. 2.Such relations as(iii)and(iii)'arccalledDifferential Equationx;theequation (i),which isfreefrom alldifferential coefficients, iscalled asolution of(iii). As,inpassingfrom(i) to(iii),asingle arbitraryconstant, wasremoved, soconversely, inpassingfrom(iii)to(i),itisjusttoexpectthatasingle arbitraryconstant willbeintroduced; and since;, ineliminatingn arbitrary constants, there areneeded theequations givingthe firstndifferential coefficients inaddition totheoriginal equation, soconversely,inpassingfrom sucharelation between differen- tialcoefficients uptothewthinclusive toanequationfreefrom them andequivalenttothis relation,itistoboexpectedthatn arbitraryconstants willbeintroduced. 3.Itisnotdifficult toseehowthesearbitrary quantitiesmust outer intothesolution ofthe(.'([nation. Forthesake ofsimplicity letusconsider anequationsuch as inwhichMandNarefunctions ofxand//.Let //;andyrepresent theCartesian coordinates ofapointPinaplanereferred totwo rectangularaxes;then theequation (i)istheequationofacurve, and /isthetrigonometrical tangentoftheangle, which the tangenttothecurve atthepointPmakes with theaxis of#, sothattheabove differentialequation givesthedirection ofaline atevery pointintheplant!.FxtunypointAbetaken onthe axis ofy,and lotUHproceedfromAforaveryshort distance inthedirectiongiven bythevalue of ':'which ithasatA;we shall thuscome toanotherpointJi.Letusproceed nowfrom Ji throughaveryshort distance inthedirectiongiven bythevalue of .which ithasatH;w<;shull thuscome toanotherpointC. Ifthisprocess bocarried out foranumber ofdirections insuc- cession, afigurewillbetraced intheplane ;and,when each of thedistances through which wesupposethetracing pointto 4 INTRODUCTION. [3 passing through A.This curve willhave adefiniteequation , which maybeexhibited intheform whereyistheordinate ofA.Hadanother initialpointA'been chosen instead ofA,thenanother curve would havebeen obtained andinto itsequationthemagnitudeoftheordinate ofA1would have entered; thesame result would ensue fromtakingieachpoint insuccession ontheaxis ofy,becausegenerallyonecurve andonly onepasses througheach suchpoint. Aseachequation,orone single equationastherepresentativeofall,maybeconsidered a solution ofthedifferentialequation,itisevident that intothe solution oftheexample wehave beenconsideringonearbitrary constant willenter;andtherefore, ifbyanymethod wecanobtain anequationfreefrom differential coefficients, itmust beexpected thatanarbitraryconstant willbecontained inthatequation. But thisarbitraryconstant obtained bythelatter method willnot necessarilybetheordinate ofthepoint,atwhich thecurve, repre- sented bythesolution, andtheaxis ofyintersect;anarbitrary element would have entered intotheequation, hadthetracingof thecurve begunfrom apointintheplanenotlyingononeof thecoordinate axes. rlni Intheexampleconsidered theequation giving-~hadonly a.singleroot;when itisoftheform then theintegral equationwillbeoftheform whereAisanarbitrary constant. And itisnot difficult tosee- that, ifthedifferentialequation beofthenthdegreein-/,then CuX thecorresponding integral equationwill contain anarbitrary constant raised tothenthandlowerpowers. 4.From -what hasbeen said astooneofthemethodsby which differentialequationscanbeconstructed, itmightbedeemed aneasymatter toreturn from the differential totheintegral eauation :butthis isnot SO. Thfiatms nfan 4.] INTRODUCTION. 5 beretraced, andtherefore some other method ormethods must be adopted. Themethods which aremost effective forthesolution ofseveral different forms ofdifferentialequationswillbediscussed hereafter. 5.When wepassfromagiven integralfunction totheequi- valent differentialequation,thelattermayprovetobeofaform which isnotincluded amongthosealready known;soconversely, ifwepassfromagivendifferentialequation, wemust notexpect toarrivenecessarilyatafunction which willbeincluded among those, with thepropertiesofwhich weareacquainted.Itis 'therefore desirable toindicate what, insuch acase,would be meant bythesolution ofthedifferentialequation. When, inalgebra, weaskwhether anyparticular equationcan besolved, wethereby enquire whether thevalue ofthevariable, which occurs init,canbeexpressedinterms ofknown functions. Thus, forinstance, intheequation ax=b thevalue ofxcanbeobtainedimmediately byaprocessofdivision: But lettheequation be P-JT. Tosolve thiswehave tointroduce afunction, which wasnot needed fortheformerequation ;and,expressingintheform . weconsider theequationsolved. Nowequationsofthethirdand fourthdegreecanbesolved bymeans offunctionsstrictly analogous tothese thecube rootandthefourth root ofquantities;but general equationsofthefifthandhigher degreescannot besolved interms ofthese functions orcombinations ofthese with similar functions. Itdoes nottherefore follow that solutions ofthese equationsdonotexist;theycanonlybesolved when functions, unused inthesolution ofequationsoflowerdegrees,areintro- duced.^ Similarly,inthecase ofadifferentialequation,whenwesay that itcanbesolved, wedonotmean toimplythat thesolution must beexpressibleinterms ofpurely algebraical functions, of 6 INTRODUCTION.[5. exponentials (includingsines andcosines), and oflogarithmic functions(includinginverse circularfunctions). Theequation dx isequivalentto Butsupposethat thepropertiesofthelogarithm were un- known, andthatthedifferentialequation dy^l dxx wereproposedforsolution. Weshould thenhave and, calling [dx x weshouldprovetherelation andbecomeacquaintedwith thepropertiesofthisnewfunction so asto'include itamongst known functions. But,hadwenotbeen able todeduce theproperties of/ (x),thevalue ofygiven by, Cdxt would stillhave been considered asolution ofthe differential .equation.Infactevery differential equationisconsidered as solved, when thevalueofthedependent variable isexpressed asa function oftheindependentvariablebymeans either ofknown functionsorofintegrals,wlieth&r theintegrationsinthelatter canorcannot beexpressedintermsoffunctions already known. Thus, forinstance, ,dx .'% isasolution of x^=<f althoughthevalue ofycannot beexpressedotherwise than inthis formwithout theintroduction ofanewfunction theproperties of 5.]INTRODUCTION. 7 which canbeinvestigated.Inthiswaythesolution ofdifferential equationsiscontinually suggesting newfunctions tobeadded to thestock ofthosealreadyknown. 6.Before -weproceed farther, itisdesirable togivedefinitions ofsome terms used inthesubject. Any equationwhichexpressesarelation betweendependent variables, their differential coefficients ofanyorder whatever, and theindependentvariables iscalled adifferential equation. Differential equationsaredivided intotwospecies,viz. : I.Ordinary differential equations,intowhichonlyasingle independentvariable enters, eitherexplicitlyorimplicitly,and tothis variable allthe differential coefficients have reference. Should there beseveraldependent variables, thenumber of equations necessaryfortheircomplete determination asfunctions oftheindependentvariable isequaltothenumber ofsuch variables. Thus, forinstance, wemight have inwhich a?isafunction oftheonlyindependentvariable t;and inwhich xandyareboth functions of t.^ II.Partialdifferential equations,intowhich twoindependent variables atleastandpartialdifferential coefficients withregard toanyorallofthese variables mayenter. Ifseveral dependent variables bepresent,thenumber ofseparate equationsmust be thesame asthenumber oftheseparate dependent variables; buttheoccurrence ofsuchsystemsofequationsisrelativelyrare. Asexamplesofpartialdifferentialequations wemayconsider daffy 8 INTEODUCTION. , tyand o= Theorder ofadifferentialequationisthesame astheorder ( thehighestdifferential coefficient itcontains. Thedegreeisthepowertowhich thathighestdifferentia coefficient israised, when theequationisinarational forman freed from fractions. Theequation -VJL dx isofthe firstorderandseconddegree ;theequation isofthesecond orderandseconddegree. Ifadifferentialequationbesuch that,when itisrationalise and freed from fractions, the differential coefficients andth dependentvariable enter inthe firstpowerandthere aren productsofthese, while thecoefficients intheseparateterms ar either constants orfunctions oftheindependent variables, th equationiscalled linear. Thefollowingareexamplesoflinea equations: The relation, which exists between thevariables themselve without their differential coefficients andwhich isthemostgenera onepossible,iscalled sometimes thegeneral solution, andsome times theprimitive,ofthedifferentialequation. 7.Theprocessofderivingtheprimitive from agivendii ferentialequationwillfrequently bethededuction ofafirs L. 7.] INTRODUCTION. 9 integralofthe differentialequation,that is,anequationofan order lower byunitythan that oftheoriginal equation and containinganarbitraryconstant;then ofafirstintegralofthe latter which willbeasecondintegraloftheoriginal equation; and soon,until differential coefficients cease toappear.This willbethecasewhen theoperationhasbeenrepeatedthe number oftimesequaltotheorder oftheoriginaldifferential equation. Now theform ofthe firstintegralwillbeaffected byanytransformation towhich theequation maybesubjected priortointegration; and, since agiven equation maybetrans- formed inanumber ofdifferentways,there willbeacorrespond- ingnumber ofdifferent firstintegrals.Butthese willnot allbe necessarily independent;and, asamattey offact,iftheequation "beofthen^order, itcannot havemore thannindependent first integrals.Forexample,thedifferentialequation hasthefollowingfirstintegrals,viz. : -jcosx4-ysinx=5, (LCfj dy. ~*sinx+ycos ac=C, dy^=yo>t(* +); buttheyarenot allindependent,thefour constants A,B,C,a beingconnected bytheequations B=Acosa, C=AsinCL When asystemoffirstintegralshasbeen soobtained inany case, itcanbeused asasimultaneoussystem,from which the highestdifferential coefficients canbeeliminated; and ifinde- pendentfirstintegralsoftheequation, equalinnumber tothe order oftheequation,have been obtained, all^thedifferential coefficients canbeeliminated fromthem soastoleave theprimi- tive. Thus fromthesecond andthirdintegralsintheforegoing example wemightdeduce y=Bsinx+cosx, 10 INTRODUCTION*[7. andfromthe firstandfourth y=Asin(x+a),. eachbeingaprimitive ;these solutions areseen tocoincide on account oftherelations between theconstants. 8.Weproceed now togivereasons forthestatement made inthelastparagraph. Adifferential equation oftheordernhas n,andcannot have .more than n,independent first integrals. From what hasalready been said itisclear thatanintegral relation between yand scinvolvingnarbitrary independentcon- stants would lead toadifferentialequationoftheorder n.Let thegiven integral equation bedifferentiated n 1times in succession;then 1resulting equationswill involve allthe differential coefficients uptothe(n l)tbinclusive andthere will, with theoriginal equation, benequationsinall.Now fromn equations,inwhich nquantities occur, allbutoneofthesequantities canbeeliminated. Letthenarbitraryconstants bedenoted by Cj,C72, ,Cn;andfrom thenequations,which wehave, letus eliminate allthearbitraryconstants except CrTheresulting equationwillinvolve thevariables andthederivatives ofyupto the(n-I)111inclusive and willalsoinvolveC^;itwilltherefore be afirstintegralofthe differential equationoftheordernwhich is equivalenttothegiven integralrelation. Now eliminate allthe arbitraryconstantsexcept<72;theresulting equationwillnow involve <72and, asbefore, derivatives ofyuptothe(n-l)thin- clusive and willtherefore beafirstintegralofthe differential equation;itwill,moreover, beindependentoftheformer, since (72 isindependentofCf 1.Proceedinginthiswaywith alltheconstants inturn,weshall obtain nindependentfirstintegrals,each ofwhich arises from theelimination ofallbutoneofthenindependent constants. Asthere arenotmore thannindependentconstants,occurring inthegeneral integral equation, anyother constant, which could appearinit,must dependon19(7a, ,Gn\letAbesuch aconstant, and lettherelation between them bedenotedbythe equation INTRODUCTION. 11 'henbetween this,andtheoriginal integral equation, andthe 1equationsobtainedbydifferentiation, (formingn4-1equa- .onsinall),thenconstants Gmaybeeliminated andtheresult illinvolve thedifferential coefficients uptothe(n l)thinclu- iveandtheconstant A.Thiswould beafirstintegralofthe ifferential equation,but itisnotindependentofthenalready btained;forfrom these lettherespectivevalues ofthequantities 'interms ofthevariables andthedifferential coefficients ofy ederived from theseparate equations,inwhichthey occursingly ndbesubstituted intheequation -^= ;thisequationwillthen eoneinvolvingthedifferential coefficients uptothe(n l)tiland lieconstant A,and willtherefore bethesame astheforegoing, nfactthetwoprocessesaremerelydifferent methods ofobtaining heone result, andthesecond shews that the firstintegralso btained isderivable from theothernfirstintegrals. Hence the ifferential equationoforder nhasnotmore thannindependent .rstintegrals.. 9. .Itisconvenient toaddheretwolemmas towhichfrequent inference willsubsequently bomade. LEMMA I.Lotultu9, ,unbenfunctions ofthenvariables x>ara, ,#,those variablesbeing independentofoneanother; Famongthose; functionsanyrelation, whichmayborepresented 'by ^(X,V,t*J=0(i), >eidentically satisfied, sothatz^,wa, ,unarenotindependent foneanother, then theequation " ' =0(ii) a/-; widenticallysatisfied. Sinceu({uation (i)isidentically satisfied, when forul9ut, ,un tresubstituted their values interms oftheindependent variables, ,hepartialdifferential coefficients ofFofthe first order with egardtoeach ofthese variables areseparatelyzero. Thus we lave 12 INTRODUCTION. [9 3!^9-l+^^2 + ..[^FdMn=Q oF oUt*oFuu^ oFou A dFdu.dFdu dFdu ,~ i.-j-s-(- -|-:=(Jm ouox c)u.ox Buox Lettheratios ofthenpartialdifferential coefficients ofFwith regardtotheusbeeliminated between these n,equations,which arelinear inthesequantities ;theresult oftheelimination is ^U l^U2 ^"u_f\ dx* d.v''3#. 11 i I and this isidenticallysatisfied. The value ofadeterminant is unalteredbythechangeofrows intocolumns andcolumns into rows; when thesechangestakepljicothoaboveequation becomes equation (ii),which istheroforoidenticallysatisfied. LEMMA II.Theconverse ofthis isalsotrue: IfM,,?/k, ,nn benfunctions ofnindependentvariables#,,#a, ,xn,and if theequation lf>Li"7 /i^J// iQ ^3"afin9 j beidentically satisfied, then thufunctionn nltu^,//wurinot independentofoneanother, butareconnectedbyareflation ofthe form 9.]INTRODUCTION. 13 Ifthen1functions uvuz,.........,un_^benotindependent ofoneanother thenthepropositiontobeprovedisatoncegrafted; wemaytherefore suppose themindependentofoneanother. Between thenfunctions uwecaneliminate nIofthe variables;iftheremaining variable, sayxn,benotthereby eliminated theresultmaytiewritten intheform H=0(Mi, ,,...,ttM,aJ. Iftheequationofcondition bewritten intheform' wemaywrite thetheorem forthemultiplicationofdeterminants intheform The left-hand side iszerobyhypothesis.Since thefunctions ul9u2,......,un_rareindependent,the first factor ontheright- hand side isJ,.aiid thesecond is^'^'""u -^..One ofdxn 3(a? lf0?2J...,<O these must therefore vanish. Ifitbetheformer, then <f>isex- plicitly independentoficnlsothatunisafunction ofM X,u2,...,ww_1 only;andthere isthusarelation between theoriginalnfunctions. Ifitbethelatterwehave anequation,whichcorrespondstothegiven equationofcondition butinwhich there areonlyn1functions ofnIvariables, since forthedifferentiations thatnowoccur#maybeconsidered aconstant. This istreated inthesamemanner asbefore;andwe should findeither that there isarelation between ul9u2,...,un-1 considered anfunctions ofxltxs,......,scn_vorthatanewequation ofcondition involvingn2functions ofn2variables would hold. Iftherelation between ul9uz,...,wn-1exist,itwillbeofthe form *("!>>......>w*-i>*J=o; which willinvolve scnsincewehaveassumed thatul9u^ ...,un^ areindependentof(meanother. Betweenty=andun=<pwecan eliminate xnandobtain arelation between ul7wa,...,un. 14 INTRODUCTION.[9. Proceedinginthismanner anddiminishing byunityeach time, thenumber offunctions, which enter intotheequationof condition, wecanprovethatoneofthetwonecessary inferences ateach reduction isthestatement contained intheproposition. Andwhen thereduction hasbeenrepeatedn1times theonly alternative ofthepropositionisthatanyfunction, chosen atwill, ^ should besuch astosatisfy=forsome variable xwhichcan bechosen atwill. Asthis isevidentlynotthecase,thetruth of thepropositionfollows. 10.Asaparticularcase ofthegeneral lemmas wehavethe following.LetUandVbetwofunctions oftwoindependent variables xandy;then ifVcanbeexpressedasafunction ofZ7 alone,wemust have _ = dxdy dydx' andconversely,ifthisequationbesatisfied, then there isarelation between UandVsatisfied forallvalues whatever ofxandysuch that Ex. 1.Arethefunctions x+2y+z, #-2y+3z, 2#3/ independentofoneanother 1 Theequationofcondition is 1,1,fy-z 1=0, 2,-2,2#+4s 1, 3,-#+4y-4zI which isevidentlysatisfied since 3rdrow= 2(1strow)-(2ndrow) ; andtherefore thefunctions aredependent. Tofindtherelation between them,ifwecallthem iilyw2>ws>wehave andtherefore 4uz=u^2-ii^ onsubstituting these values'. Ex.2.Prove thatthefunctions axz+byz+cz*,Ax+By+Cz, and oW(#c+C*6) +6y(&a+A*c) +cW(A*b+B*a)-2oic(BCyz+CAzx +ABxy), arenotindependent ;andfindtherelation between them. CHAPTER II. DlFFEKKNTIAL EQUATIONS OFTHEFlRST ORDEE. 11. TIIJEgeneraldifferentialequation ofthe firstordermaybe resented by F ireFisarational andalgebraical function sofarasthe differ- ialcoefficient isconcerned, [nthisgeneral form theequation not l>eintegrated;but tinTOarecertainparticular forms, to orother ofwhich many equations canbereduced, andwhich litofimmediate, solution. These forms wemaycallstandard ns. 12.Hefort?considering them indetail, wewillproveapro- it.ion, whi<:h ismerelyaparticularcaseofthegeneraltheorem icai-ed in 8,viz.,that adifferentialequation expressiblein form ireMandNare.one-valued functions ofxandy,canhaveonly independent priiiLitive. Supposu that, ifitbepossible,twoprimitives ob!n obtained. From the firstofthese the"value of~isax L-nby *. ,S(M#_o^,Tn 17~~u> 18 EQUATIONS OFTHEFIRST OBDER. 14.STANDARD II.Linear Form. When theequationofthe first order islinear, itmaybe 1written intheform wherePandQarefunctions ofxandareexplicitly independent of y.Multiplyeach sideby </**>; then, since ' * theequationbecomes onintegration (theleftside isnowaperfect differential) weobtain astheprimitive thatis, Asinthegeneral case, T(xdx hence Ex. 2.Solve(i) ^(ii) 14]STANDARD FORMS. Ex. 3.Shew thattho solution ofthegeneral ocjuntiim mayboexhibited intheform 15.Animportantassocuited form, which canbesolvedbythe same method,is wherePandQarcfunctions of .7;alone;. Divide byyn ;theequationthon is __l *(L\..i 7Zr1<LX or/7""(" which isthestandard form;andthogeneral/solution is ir*e =,-(?/- ,/,. Ex. 4.Solvu ThiHIJUCOIIIUH,aftciratnni.sfornintinii wimilar totho ^i\I 1 I, theprimitiveofwhich in ThiH inic"^,-^- f^1(JKf,-lv ^ j. I,,p/.i'I<iKj;V J.^""f whence Ex. 6.Solve(i; (> (' (i) |J; (iv)5'I_ .'/" formEQUATIONSOFTHEFIRST ORDER. Ex. 6.Shew thatthefourequationsin7leadtothesameprimitive. 1$STANDARDIII- Homogeneous Equations. theequationwhen ofthe firstdegree andexpressed inthe said tobehomogeneous,whenMandNarehomogeneous actions of*andyofthesamedegree. Inthis casewecan functions write rheingthedegreeofMand jST.Onthesubstitution of y=tw, sothat vmaybeconsidered anewdependent variable, theequation becomes dx<f>(v)dv _01xv(>v-~' inwhich thevariables areseparated ;theintegralis Theprimitivewillbegiven bythesubstitution of-forv CG after theintegrationhasbeenperformed. Iftheequation however benotofthe firstdegreebut still homogeneousinxandy,itmaybewritten intheform F\y,41-a \x'dx). . .. There arenowtwomethods ofproceeding.The firstmethod -5- ; istosolve thedtyuation consideringitas-anequationin-5- ;leta solution beexpressed by }gl KTANDAltD FOHMS. This isthe cast-alreadydiseuswd. Thesecond n.ethod wUwdvi* th*1njimtinn rmiMdiTintfit- anequationin^;thruweshould h:iw or y-'!</') wherepwwritten f<*':'(. DifliTnitintinj;thiswith n-^inll wehave ?'-/,(!') +<(/')' andthcrefon* rir^/' */' Thisgivesonintegration say;theeliminationof/;!ji*tw<M*n tin* hi^f // willgivetheprhnztivr. Hut itisn)(U!WU\H clrsimliN* t p\itmayberftuini'fl imtin-|fiir;iiii<t<'r<!'ajiniufittin-*rirr" Bponding curve, inwhSHifjw itsuwwmild }>-Mniilurt flu-it. <f theeccentricnngleofujHaui.onani-Ilipw, Ex. I.Hwlvv r.f.v';v ,-tfv.r/.c When wewrity^/v,thifjiiutinii luTMtn*^ whence or 22KQI'ATIONS OKTHKFIRST ORDER. Kr.S. S,,hv.(i)s+ff'-ny., ./-A+(mvl//:/-+7,andsuppose hmid sochosen that ihi* IfIjriwuviT '^ ^I)llt f;f^tt*ll>Hfr<" each ofthesefractions, then <t|itJLf.i"!i!* ^ivinx//and /an*inccniHiHtont. Leteach oftheequal ra In-ffpinlt.u //(;tlli:li <//,/+ty+;-=?;i(a*+ty)+a Suhht ituii5 ft.t-+/*//= ; .. ,,'* UH-II "+/ HIK!thnvarint'lcH arcKcparahlc. If, //,tin* c;iillation in tth <: )that //-? A>. -1.Scilv(i)fy-"/+7^=(3.?:-7y-3;^; (ii) (iii) Af .i:.T.Shew thatthe inwJiich /',VniidVi arnhotiiognnooiwfunotions ofxandy,PandRbe ofthowiinu decree, ty^Holvwi bytheHubstitution y==vo?. A>, IJ,Hiilvo +^ 17.]STANDARD FORMS. 28 17.Letnow thecurves, whoseequationsarcthecomplete primitivesofthehomogeneous equation,betraced; they fqnna systemofsimilar curves. For letthere hedrawn through the origin anyradius vectorcuttingallthese curves andmaking an angle6with theaxis ofx;theinclination totheaxis ofxof thetangenttooneofthecurves atthepointwhere thin vector meets itisgivenby and therefore allthetangentsatpoints lyingonthis lineare parallel. And therefore thecurves are allsimilar andsimilarly situated. 18.STANDARD IV. Equationsarise inwhich oneofthetwovariables doesnot explicitlyoccur. Consider firstthat class fromwhich theindependent variable isabsent. Theequationwillthenbeoftheform Asinthegeneral equationunder StandardIII,,there aretwo th thatmethods ofproceeding.Ifitbepossible, wernayHO!VO fory-so inwhich thevariable's aroseparable ;thoprimitives w [dy J/fody . Or,ifitbopossible, wemaysolve fory;wippoHo uHohilion to begiven by -/.-/.. Differentiatingwithrespecttoxwohavo 24 EQUATIONS OFTHEFIRST OBDER.[18. inwhich thevariables areseparable:andtheintegralis which, when combined with fortheelimination ofp,willfurnish theprimitive. Itmaybe more convenient toleavepuneliminated. Letusnowconsider theclassfromwhich thedependentvariable isabsent. Theequationwillthenbeoftheform 1 da;. Since T--j- dosdy thisequation maybewritten anequationoftheformerclass,andsolublebythemethods thereto applying.These methods however maybeapplied totheequa- tionwithoutmakingitundergothistransformation.Solvingthe equationifpossiblefor--,weshallhave andtheprimitiveistherefore fit]Orsolvingforxinterms of-/-,when this ispossible, weshall obtain Differentiating withrespecttoy(theabsent variable') wehave l-=F'(v)*Ppl(P)dy' 18.] STANDARD FORMS. 25 theintegralofwhich is y=fpF 1'(p)dp+C. This,combined with a;=F1(p), constitutes theprimitive. ** Solve(i),-.J+5(|)8 ; 19.STANDARD V.' When theequationofthefirstorder isofthentb degree, itarrangedindescending powersofthedifferential coefficient, m that itmaybewritten inwhichPlfPv......,Pndenote functions ofaoandy.Ifwelook uponthis asanalgebraical equationin-,which hasnroots pvpa,......,pn(these beingfunctions ofCDandy\theequation becomes dy This canbetrueonly,ifoneormore ofthefactors ontheleft- hand sidevanish;andthereforeanyrelation between asandy, which makes afactor vanish, willbea-solution oftheoriginal equation,while norelation which doesnotmake some factor vanish canbeasolution.Supposethen thattheprimitivesoftheequa- tions (deduced bymeans ofoneorother ofthepreceding methods)are respectively;allpossiblesolutions ofthegiven equationwillbe contained in 26 EQUATIONS OFTHEFIRST ORDER.[19. Butthegeneralityofi\isintegralwill stillbemaintained, ifall theconstants C ltGv......,Onbemade thesame, sayC;forinorder tofindavalue ofywemustequatetozerosome factor onthe left-hand sideofthenewform, andthiswouldgiveanequationof theform NowGisanarbitraryconstant;ifthen allpossiblenumerical values begiventoit,theremust beincluded intheseries ofcon- sequent equationsalltheintegrals,which canbederivedsimilarly from thecorrespondingfactor ofthe firstproduct. Hence wehave asthegeneral complete primitiveoftheoriginaldifferential equation <k(a,y,0)& (as,y,C)............ <(a,,y,0)=0. Ex. 1.x*p* Then xp-y which, bythesubstitution y=xz, becomes _ (!+/)* When thepositive signistaken, thesolution is Thenegative signgives=ainh(c-x); hence thegeneral solution is A*. Solve(i)(!)'-?=0; Ex. 3.Solve (i) (ii) (iii) (iv) (v) 19.]STANDARD FORMS."27 (vii) JJr -#. 4.Shew that,ifthegeneral equation behomogeneousin#'andy,it canbesolved bythesubstitutions dt y-te,*=*. Hence solve 20.STANDARD VI. Clairaufa Form. Theequationtowhich thisname isusually appliedis inwhichpstands for-j-. CuOB Differentiate theequationwithregardtox :then sothat either dw or a>+f(p}=0. Takingthe firstofthese,wehavep=caconstant;andhence the primitiveis y-ca+f(c). Thesecondequation expressesa;asafunction ofp,andtherefore ifpbeeliminated between thisequationand &relation between yand aswillbeobtained. Ofthese theformer isevidentlyasolution oftheequation,and from itthedifferential equationcanbededuced atonce;foron differentiating weobtain p=c, andeliminatingcwehave y=pac+f(p). 28 EQUATIONS OFTHEFIRST ORDER.[20. Ifnowweturn totheother relation between 00andy,which willbethatderived fromtheelimination ofpbetween y=px+f(p)\ # itisatonceevident that itcontains noarbitraryconstant andsois notageneralsolution. Tet itmaybeasolution oftheequation ; fordifferentiatingthe firstequationwehave =P bythesecond equationunless^rbeinfinite; eliminating^)fromthe j equations y=pa;+f(p)and -r-=pweobtain which istheoriginal equation. 21.Therelation between thetwosolutions, when both exist,is easilyindicated bygeometricalconsiderations. The firstsolution y=co)+f(c) representsafamilyofstraightlines;iftheyhaveanenvelope,itis found bydifferentiatingtheequationwithrespecttoo(infact, this isequivalenttogivingcapairofequalvalues forthesame values ofxandy)andthenwehave 0-*+/(c> Theresult oftheelimination ofcbetween theseequationswillbe thesame asthat ofeliminating pbetween thetwo y=pas+f(p), andtherefore thecurverepresented bythelatter istheenvelope ofthefamilyoflinesrepresented bythe first solution, should these lineshaveanenvelope. Such asolution oftheequation, which isnotincluded inthe primitive (butwhichmaybederived from itintheabovemanner), iscalled aSingularSolution. Weshallshortlyreturn toamore detailed discussion ofsingularsolutions. *Itshould benoticed that forpurposes ofelimination pismerely aquantity likelytodepend uponyanda;itisnotnow necessarily -^.ax 21.]- STANDABD FORMS. 29 Ex. 1.Solve y=xp+-.P The firstsolution is Thesecond isgivenbytheelimination ofpbetween andtheoriginal equation ;eliminating pwehave The latter isthesingularsolution;thecurve representedistouched byall thelines included intheprimitive. Ex.2. Solve" (i)y=*>+(I+p*fi ; (ii)y=px+p-p*; X(iii) a,yp*+(%x-b)p=y\*,.-I-^,ifc.u, ^it-*,**~J f'^"** 22.There isanextended form oftheequation, which canbe solved inasimilar manner, viz. : Tosolve this, lettheequation bedifferentiated with regardto as;then ^f(p)-p p-f(p)1 which islinear inxandcomes under Standard H. Lettheintegralbe F(tK,p,c)=Q. The result ofeliminating pbetween thisandtheoriginal equationwillbetheprimitive. Ex. 1. x+yp=ap*, or y=ap--. Differentiatingwithregardto#,wehave dp__1 ,^ y==adxpp*dx' 30 EQUATIONS OFTHEFIRST ORDER.[22. andtherefore dx x a ^ theintegral ofwhich is Thiscombined with theoriginal equationistheprimitive. The equation could alsohave been solved bydifferentiating with re- gardtoy, Ex. 2. Solve -(i)x=yp+apa ; Xii)y=orp+ax(1+p^ ; (iii)y= (v) SINGULAR SOLUTIONS. 23.From theinvestigation,of21itisclear thatasolution of adifferentialequationcansometimes befound, whichjs no^incjudsd inthejprimitive;suchasolution doesnotinvolve initsexpression anyarbitraryconstant. The limitation ofnotbeingincluded in theprimitiveismostimportant;forinthelatter aparticular value, sayzero,could beassignedtothearbitrary constant, andthen asolution would befurnished butnotofthenature indicated. Weproceed now toconsider thetheoryoftheseSingular Solutions ofthegeneraldifferentialequationofthe first order, which willbewritten~" <t>(>y>p)=Q- Ifthedifferentialequationeither belinear orberesoluble intoasetofrational linearequations (asinthecaseof"Standard V.)then ithasnosingularsolution;anysolution ofitapparently ofthisnature ismerelyaparticularsolution derived from, the primitive bygivingaparticularvalue tothearbitrary constant therein contained. Forthepresent purposetherefore theequation inpmaybeconsidered irresoluble :ifitcanberesolved into factors which arenotlinear andnotresoluble into linearfactors, thenweshould consider inturn each ofthese irresoluble factora 23.] SINGULAR SOLUTIONS. 31 Wemaythus consider<_=asarational andirresoluble equa- ^ tionofdegreen.Moreover weshallassume that isaone-valued-1 function, andthat itcontains nofactor, which isindependentofp\1 /> such afactor,ifitwere retained andequatedtozero,wouldsatisfy theequation,butwould notinvolve thedifferential coefficient. Ifinanycase these factors occurred, weshould supposethem removed. 24.The considerations adduced intheIntroduction famish theinference that, ifaandybethecoordinates ofapointin aplane, the differentialequationdetermines asystemofcurves inthatplane,whichdepend uponasingle independentvariable parameter; and asthedifferential equationdetermines atany point adirection throughthatpoint,there willbendirections, given bythevalues ofpthere, andtherefore ncurves willpass through anypointintheplane. Torepresentthissystem alge- braically weneedanequationoftheform f(as, y,c15cs,,0= '- which isrational andalgebraicalandtheconstants inwhich are also rational andalgebraical;butasonlyasingle independent parameterisneeded, there willbeamongthese inconstants in 1 algebraicalrelations. Further thisfunction /willbeone-valued; andanyfactor, involvingasandy(oreither ofthem) butnone of theconstants, would berejectedforthesame reason asledtothe rejectionofsimilar factors from thedifferential equation.Asthe differentialequationcannot beresolved into simpler equations ofalowerdegree,thealgebraical equationisnotsoresoluble;ifit were, toeachalgebraical equationoflowerdegreethere would be acorrespondingdifferentialequationoflowerdegreearesult excluded byhypothesis. And thereason thatmconstants con- nected bym1relations areinserted instead ofasingleconstant isthis; theequationinthe latter casewould bethesame as that derived from theformer with alltheconstants eliminated except one,andasthiselimination would usually imply operations (suchassquaring, &c.)which introduce equationsother than that wanted, theresult would bethatthefinalequationVouldrepresent more thanthesingle equationdesired. Forexample, supposethat byanyprocessanintegralisobtained intheform [a?+y*-a(ascosa+ysina)}9=a" 32 EQUATIONS OFTHEFIRST ORDER.[24. orchangingtoalgebraicalconstants [af+tf-afa with thecondition then theequivalent equation containingoneofthese constants, asm,alonewouldrepresentnotonlythisequation hutalso {a?+f-a(-Ix+my)}a=a8 (a*+y\ with thesamelimiting condition, andtherefore would notbe equivalent solelytothe firstofthese. Further wehavencurvespassing through every pointinthe plane ;hence theequation/= 0,withthem Iequations between theconstants, mustgiveatevery pointnsets ofvalues forthese constants. Lettheaggregateoftheconstants bedenotedbyC,so that foranypointintheplaneGwillhavenvalues. 25.Consider nowtheformation ofthedifferentialequation fromtheprimitive ItisobtainedhpeHminatingtheconstants between them 1 relations, thisequationandtheequation + = dadyda Butsupposethequantities Creplaced byfunctions of#;the deduction ofthedifferentialequationwillbethesame asbefore, exceptthat forthelastequation wemust substitute dxdydoc Theresult willbeactuallythesame asbefore, if Tosatisfythisequation wemust have either -=-zerowhichdx leavesGconstant ;or(?TnnHt. bedeterminedby AMAN RE8EAROH INSTITUTE BANSALOK 6 25.] SINGULAR SOLUTIONS. 33 Letthevalue ofCsodetermined besubstituted inthefunction/. Wemaythus ingeneralasasolution ofthesame differential equation equatetozerothediscriminant of/withregardto(7; letthisbewritten 26.This locus isthelocus ofallpointsintheplaneatwhich theparametric constants Chavetwoormoreequal values; and initthere willtherefore beincluded ^ (i)thelocus ofallthenodalpoints (double, treble, etc.)of thesystemofcurves;foratsuchapointthere areasmanyvalues ofCequaltoeach other asthere arebranchesthroughthepoint, since thebranchesbelongtothesame curve; (ii) thelocus ofallthecuspsofthesystem,forsimilar reasons; (iii) theenvelopeofthesystemofcurves, whichmaybeeither asinglecurve orseveral;foranypointontheenvelope maybe considered asbelongingtotwoseparatebutconsecutive curves of thesystem,theconstants ofthese consecutive curvesbeingulti- mately equal. [Inthecase,when theenvelope"Qanbedecomposed into several curves,itmayhappenthatoneofthese ismerelya particularcurve ofthesystem f(a, y,C)= ;itsequation might beexcluded asbeingaparticular solution.] Letthese threerespectivelybecalled thenodal locus, thecus- pidal locus, andtheenvelopelocus. 27. Ifwenowconsider thedifferentialequation toV>P)= inconnection withthesystemofcurves, whoseequationconstitutes itsgeneral solution, itisevident that theenvelope ofthesystemisa solution oftheequation;foratanypointontheenvelope (which isapointontwoconsecutive curves)thedirection ofthetangent isthesame asthat ofthetangenttoeither ofthese ^curves atthat point;andsince thedifferential equationissatisfied bythequan- tities, which areconnected withtheelement ofthesystemofcurves', itmust besatisfied bythese(unaltered) quantities,-which 'are"coh^* nected with theelement oftheenvelope.'..... F. 3 34 EQUATIONS OFTHEFIRST ORDER. [27. Butthenodal locus isnotasolution oftheequation;ifitwere, thedifferential equation would, forthevalues ofxandyatany node, besatisfied bythecorrespondingvalue ofpatthispoint onthenodal locus. Rememberingthatthenodal locus isformed byaseries ofpointsonoursystemofcurves, weknow that the values ofpatanysuchpointwhichsatisfythe differential equationarethosegiven bythatcurve ofthesystem which passes throughthepoint.Butasthetangenttothenodal locus atsuch apointwillnotingeneralbeatangenttoanyofthebranches of thecurve ofthesystematthepoint,itfollows thatthevalue ofp forthenodal locus differs from those values ofpforthecurve of thesystemwhichsatisfytheequation when substituted initwith thecoordinates ofthepoint. And itwouldonlybebyaccident thatthevalue ofpforthenodal locus could coincide withanyof theremainingvalues ofp,which donotbelongtothecurve on which thenode lies,butarefurnished byother curves ofthe system throughthatpoint. Hence thevalue ofpforthenodal locus atthepointwillbesuch asnottosatisfythedifferential equation ;andthenodal locus willthereforenotbeasolution ofthe differential equation. Exactlysimilar considerationsappliedtothecuspidallocus lead toasimilar conclusion :thecuspidallocus isnotasolution of thedifferential equation. 28.Now theenvelopeofthesystemcanbederived from a knowledgeofthedifferentialequation alone, Le.without aknow- ledgeoftheprimitive. Atanypoint ontheenvelopeatleast twoofthebranches ofthedifferent curves coincide indirection; and therefore forsuch apoint weshallhaveequalvalues ofp belongingtodifferentbutconsecutive curves. Ifnowweerpressthecondition thattwovalues ofpshallbe equal, bymeans oftheequation andeliminate pbetween thisandtheoriginaldifferentialequation (infact,equatethediscriminant of <tozero), then thelocus Diactp(a:,y,p)=Q willbeoneatpoints alongwhich twovalues ofpwillbeequal, andwillobviouslyinclude theenvelope. 28.] SDIGULAB. SOLUTIONS. 35 Butbesidesincludingtheenvelopethisequationwillalsogive thelocus ofallpoints (i)atwhich twobranches ofthesame curve touch,i.e.will* giveallthecusps;thistherefore asBefore isthecuspidallocus. (ii)atwhich twocurves whiclijg[ta_differfint butnotconsecu-| tivetouch; thislocuTs~called atqc-locus. Thus, forinstance, if wehavetwoinfinite series ofconcentric circles oneround each of twopoints,thestraightlinejoiningthecentres (and produced bothways)isthelocus ofpointsofcontact oftwo circles, one belongingtoeachsystem. Asbefore thecuspidallocus isrejected,notbeingasolution; andreasoning exactlysimilar tothatwhich ledtotherejectionof thenodal locus indicates thatthetac-locus isnotasolution. 29.Hence ofallthese theonlysolution ofthedifferential equationistheenvelope-locus; and this,and this alone, wecall the" SingularSolution"ofthe differentialequation. ,Either method ofobtainingtheenvelope-locus mayintroduce some of theother lociwhich havejustbeenshewn nottobesolutions; and therefore inanyparticular case, unless theequation derived obviously representstheenvelope andnothingbuttheenvelope, itisnecessarytotrywhether theresult satisfies thedifferential equation. Should itnotdoso,itmayhappen thattheequation canberesolved intoothers thataresimpler, andoneormore than oneofthemmay satisfytheequation ;these willthen constitute theSingularSolution. And those which donotsatisfythe differentialequationwillbefound tobeloci,whichaccordingto theprinciplesabove explained oughttoberejected. 60. Itistobeunderstood thatanirreducible differential equationhasnotnecessarilyasingularsolution. Thus letthe discriminant withregardtopof bedenoted byU,whereUisafunction ofthevariable coefficients ofpinthisequation,andsupposethatUcannot beresolved into simplefactors. Iftheequation U=beasolution ofthedifferentialequation, then thevalue ofpisgiven by dUdU 32 30.] SINGULAH SOLUTIONS. 37 Inthecaseofeachexamplethecorresponding figure should bedrawn. Ex. I.p*y+p(ne-y)-iK=Q. Thecondition thatpshould haveequalvalues is ory=-x, which isnotasolution. Now theequation maybewritten thesolutions ofwhich are y-x=c and Thedifferent curvesrepresented areobvious. This isanexampleoftheremark(23)that,iftheequation bereducible tolinear andrationalfactors,ithasnosingular solution. Ex. 2.jDyoos2a-2pa?ysinaa+^a-icasin2a=0. -xdibit*- /~" / Thecondition thatpshould haveequalroots is -/*" *{i~fa ^* a^y*sin4a=y*cos3a(y2-a?sin8 a), ^ .jr.'. ".+>*-....' that is (^sin8a-yacos2a)yz=0,*^>t*<~*-/Zi~.-'A^/gj 8tlmt3/=0,-A^, :...__ e? and.y=a;taua.. / .t Theprimitiveis (Tv^ andthecondition that cshould haveequal roots is ory Thecurvesrepresented areaseries ofcircles;theirenvelopeisthetwo straightlinesy=+xtana,which constitute thesingular solution. Theline?/=0i8 atao-locus. ^,-j ^,. .----slo'''" t it- i'_, Thecondition thatpshould have" equal roots is Theprimitiveis J~' '' andthecondition that cshallhaveequal roots is a(xa)(x- 6)=0. The differentialequationissatisfied byx=Q,x=a, =b(and thecor- respondinginfinite values ofp);and these aresingular solutions. The remainingfactor inthej^diflcaiminant gives &i'=a+&(<z2-< andthese lines aretac-loci. 36 EQUATIONS OFTBEFIRST ORDER.[30. andwemusthave theequation 9^\ (dao\*'y'-mr 9y; identicallysatisfied forvalues ofxandyconnected byU=0.In other words, there must bearelation between thecoefficients of pin$andtheir differential coefficients withregardtoxandy; butthis willnotingeneralbethecase. Ifweconsider inparticulartheequationoftheseconddegree intheform then thesingular solution, when one exists, isS=0,where S iseitherLN-M*orafactor ofthis. IngeneralLN Al*cannot beresolved into factors;and itisnotitself asolution, unless ,/asy....asas^/asy AL=--2Jlf5-=-+JV(=-)=0, \dxj axdy \oyj whereLN=Ma ;andthese ingeneralwould betwoindependent simultaneousequations determiningxandyasindependent quan- tities. Yet,fromwhatwehave seen, theprimitiveofthe differ- entialequationisoftheform and ifthisbeanalgebraical equation,itwillhave ageneral envelopecontained in L'N'-Mft=0, which willbeasingularsolution. Theexplanationoftheap- parentcontradiction liesinthefactthat thisintegral equationis usuallyofatranscendental form, andsohasnotingeneralan envelope;andtheexceptionsinthe first casewhen the differ- entialequationhasasingularsolution aretheexceptionsinthe other when thetranscendentalequation representsasystemof curves withagenuine envelope*. Wenow-proceedtoconsider someexamplesofthegeneral theory. *Of.Oayley, Mess, ofMath. Vol. vi.pp.2387.Thetheoryofsingular solu- tions ofdifferential equations ofthe first order, asatpresent accepted, was first given byOayley intheMess,ofMath. Vol.n.(1872) pp.612. SeealsoDarboux, Bull, desSc.Math., Vol, iv.(1878), pp.158176. 38 EQUATIONS OFTHEFIRST ORDER.[30. Thecurve y*=x(x-a}(x- 6), (0<&<&)consists ofanoval cuttingtheaxis ofxattheorigin andata distance a,andofacurve likeaparabola cuttingtheaxisofxatadistance b thetangentsatallthese pointsareparalleltotheaxis ofy.Thesystem. ofcurves isobtained bymovingthiscurveparalleltotheaxisofy.The straightlines#=0,x=a,#=& areenvelopesofthesystem; theline 3as=a+b(az-ab+b'rfisatac-locus ofrealpointsofcontact, the line &c=a+b+(az-db-t-b*)*ia atac-locus ofimaginary pointsofcontact. Ex. 4.Intheforegoing make a=b;audremove(see 23)thefactor (is-a)2 ;thedifferential equationis thecondition thatpshould have equalroots is x(3z-a)a=Q. Theintegral equationis (y+o)a=tf(#-a)a , andthecondition that cshould haveequal roots is #(#-a)2=0. Common tothesewehave a'=0, which(with thecorresponding infinite value ofp)isasolution oftheequation, andtherefore asingular solution. Everycurve ofthesystem hasadoublepoint ;thelocus ofthese isx=at which isanodal locus;thelinex=%aisatac-loous. Ex. 5.Intheforegoingleta=0andremove thefactor x;thedifferential equationis 4$P=9z; thecondition thatpshouldhaveequal values is #=0. Theprimitiveis andthecondition that oshould haveequal values is ^>=0, The differentialequationisnotsatisfied by#=0(withthecorresponding in-finite value ofp). Thecurveyi=z3isthesemi-cubicalparabola having acuspattheorigin j andthesystemisobtained bymoving thecurveparalleltotheaxis ofy,so that&=Q isthelocus ofcusps, andtherefore isnotasingularsolution.- ^r. 6. . -. thecondition thatpshallhaveequal values is r^'*7-k"*''",'* Theprimitiveis /;,-|-V!j^^/ i,'v- \ A c*, cc*r\ . I_>^- 30.] MISCELLANEOUS EXAMPLES. 39 andthecondition that cshallhave equal values isobtained byeliminatinga between thisand (#-c)(j?-3c)=0, sothateither agreeing withtheformer. Both ofthesesatisfythedifferential equation ;but thefirstofthem isaparticularsolution- (correspondingtoc=0) andwethere- foreconsider thelatter alone asthesingularsolution. Ex. 7.Obtain theprimitives andthesingularsolutions (where these exist)ofthefollowing equations; andspecifythenature ofthe lociwhich arenotsolutions butwhich areobtained with thesingularsolution. (a) xpP2yp+4tf=0; Primitive aP=o(y a) ; Singular solutions y=2#. 08) (a?-aP)p*-2xyp-3?=Q; Primitive c2 -I-2cy+aa=^; Singular solution a?+y*=a2 ; Tao-locus a?=0. Primitive y=<?(a:-c}z ;'^.' i Singular solution j^-16y=0; tv" ,', Singularsolution alsoparticular.,t-^^' >f\f 2/=0.*-''*^- J (8) w (0-y*-b*)p-xy=Q; Pl*' -^)^=i;:1 Furtherexamplesoccur inthepaper byCayley,Mess, ofMath. Vol. vi.(I andinonebyJ.W.L.Glaisher, Mess, ofMath. Vol.xn.(1882) pp.114. MISCELLANEOUS EXAMPLES. 1.Solve theequations: (i)yxp'=x-^-yp\ (ii) (iii)yp+y=pz ; (iv) (v)mynxp=ypi ; (vi) (vii)z>8+JtP_=^asBp\- (^iii) (xi)y-2a;p=/(ap); (rii) ^2-?=. 40 MISCELLANEOUS EXAMPLES OF (nil) (l-.p')"-*-*^-*; (xiv) (xv) (l+6y2-3^)23=3a?ya-a;2 ;(xvi) 3/=tf (xvii) ay-\-bxp=3fmyn(cy+ezp) ; (sviii) #p(a2+y2+a2 )+x (xix) (*p-yP=pP-*Zp+l; (xi) (.tp- (xxi) (sxii) (sxiii)(a?cos^+ysiny- )y=(ysiny--xcos^ )\A- si)* Vf^ xj (xxiv) (ixv) {(a;2-ys )sina+2j?^COBa- (=2ffysina- (-y2 }cosa+A- 2.Shewthat,if where thequantities Aareconnectedbytherelation 3.Integrate theequation cos6(cos0-sin asin0)cW+cos(cos-sinasinff)<20=0. Shewthat,ifthearbitrary constant bedetermined bythecondition that theequation must besatisfiedbythevalues(0,a)of(6,0),theequation issatisfiedbyputting 0+0=a. 4.Provethat,ifthedifferentialequation eydx-(y+a,+bx)dynx(xdy-ydx)= betransformed intoanequation between uandxbythesubstitution u(y+a+bx+nxy)=y (C+TWP), thenthevariables areseparable ;andreduce theequation totheform dv_dx bythefurther substitutionv=ow+/9, aand)9being suitably determined. 5.Reduce theequation cusyp*+(3*-ay*-6)p-xy= toClairaut's fornvand hence solve theequation. Solve theequation x y 3J+V 1a hlS+~rt wherea+/3+y=0. EQUATIONS OFTHEFIRST ORDER. 41 6.Shew that,ify^andyzbesolutions oftheequation wherePandQarefunctions ofj?alone, andyi=y&then -/*,3=1+06Vl , where aisanarbitrary constant. 7.Prove thatthevariables intheequation a2 }|=y (as maybeseparated bythesubstitution x=u+v andy=fcu v,providedkbe properlychosen;andintegrate theequation. 8.Shew thattheequations arederivable fromacommonprimitive, anddetermine it. Arethepair x+p(l+py)~*=a andy-(l+ps)~^=b soderivable 1Alsothepair yp=ax and#2 (1 joa)=6? 9.Integratethedifferential equation x{ay9+(ay+bx)3 }+y-/[bo?+(ay+Ixf)=0.dx Atangent toacurve ataflypointPcutsthetangent andthenormal ata fixed pointinthepointsJ/"andNandtherectangle OMFN iscompleted. Find thecurve which issuch thatthetriangle formed bythetangentsatany three points P,Q,Risequaltothetriangle formed bythecorresponding points P',Qf,K. 10.Determine thesystemofcurves which satisfies thedifferential equa- tion andshew thatthecurve which passes throughthepointx=Qandy=ncon- tains aspartofitself theconic 11. Integratetheequation a* 7/a ^"i andexamine thenature ofthesolution 42 MISCELLANEOUS EXAMPLES. 12.Discuss thequestionwhether y=Qisaparticularsolution ora singular solution oftheequation 13.Obtain andinterprettheprimitive andthesingularsolution(ifthere beone)ofeach oftheequations (i)p%+fip3=a(y+/*#); (ii)x (iii)y(l+.p2)=2sp; (iv)i" 14Shew that ingeneralitisnecessary,fortheexistence ofasingular solution oftheequation <j>(no,y,p)=Q,thattheequations *-*|=<>>*- should hesimultaneouslysatisfied. Prove that,ifalocus ofpointsofinflexion canbeobtained from the integral familyofcurves,itwillbeincluded intheresult obtained bythe elimination ofpbetween thefirstandthird ofthese equations. Discuss thesolution oftheequation (Darboux.) 15.Obtain theprimitiveofthedifferential equation andshew thatexactlythesameequationisobtained byexpressingthe condition thatpshould haveequal roots inthedifferentialequationasby expressing thecondition that o(the arbitrary constant) should have equal roots intheprimitive ;anddetermine thegeometrical meaningofthis equation.Isitasingularsolution 1 16.Theprimitiveofthedifferentialequation iad*+c(tt+y} +l-xy=Q. Verifythisandobtain thesingular solution both from theequationinpandfromtheequationinc,explaining thegeometrical significationoftheirrelevant factors thatpresent themselves. 17.Shew thatthesolution oftheequation s Is2#=ayasingularsolution ? Trace thecurve andthelocusgiven bytheequation independent ofan arbitrary constant. (Woolsey Johnson.) 18.Shew thatthedifferentialequation which hasnosingular solution doesnotadmit ofaprimitive representing a systemofalgebraic curves. (Cay ley.) CHAPTER III. THEGENERAL LINEAR DIFFERENTIAL EQUATION WITH CONSTANT COEFFICIENTS. Preliminary 31.Beforeproceedingtothediscussion ofthelinearequation ofthe 71thorder with constant coefficients itisconvenient toformu- lateandprovecertain theorems indifferentiation andintegration, which willberequiredinthat discussion. J J3 LetDstand for-*- :D*for -=-. ,:andsoon.Then thissymboldw'da?'J Dobviouslyissubjecttothefundamental laws ofalgebra;for evidently =Du+Dv. Itisnecessarytodealwithnegativeindices;thus ifwehave Du=v and, after thealgebraical analogy, wewrite u=D~l v, wehave v=Du=D.D~1 v, sothat D .D-1=1. ThusD'1 representssuchanoperationonanyquantity that, if theoperation represented byDbesubsequently performed,the quantityisleftunaltered. Itatonce follows that these symbols withnegativeindices also follow thelaws ofalgebra;andan operationwithanegativeindex isequivalenttoanintegration. 44 THELINEAR EQUATION[31. But itisimportanttopointoutthat thespecial objectofthese inverseoperationsistofindanintegralbutnotthecomplete integral ;andthearbitraryconstant which arises inintegrationis therefore omitted. Inwhat followstydenotes afunctional symbol ;andty(as) everywheredenotes analgebraicalrational function ofccwhich can beexpandedinascendingordescending integral powers (orboth) ofthevariable. 32.Theor&m I. ForsinceDstands for-3- ax Dea*=OK?*. When each side isoperatedonwithD"1 ,theequation becomes ortransposingthesides oftheequation anddividing byawehave XT1e=<Tltf. Repeatingtheseoperations weobtain theequations Now as-^risanalgebraicalfunction which canbeexpandedin powers wemaywrite =f(a)e- 33.Theorem II.IfXdenoteanyfunction whatever ofsc,then Asingle operationwithDgives D{e'a>X\=e from which,ifboth sidesbemultiplied bye', sothattheeffect ofoperatingonXwith e"*De istogiveD+a operatingonX.Lettheoperation berepeated ;then (e-I>O (e~*De"*)X=(D+a)(D+a)X 33.] WITH CONSTANT COEFFICIENTS. 47 or(e~aaD*OX=(D+a)sX.>ofthe qf\ffit Operate againwith e~Dena :then' ("DO (e**-D9OX=(D+a)(D+a)2X* J andsoon. Iftheoperationbeperformedntimes, theresulting,-- equationwillbe whichmultiplied byenx gives Dn {ea'X}=e inwhich ndenotes apositive integer. Consider nowthecaseofnegativeindices;write (D+a)*Z=Z l sothat X=(D+a)-nX 1. Then theresultjustobtained maybewritten D"<F(D+a)~nX,=eZT. Operate oneach sidewithD~nandtheresult is Nownolimitations wereassignedtotheform ofXandthere aretherefore none onthat ofXvwhich canthusrepresent any function ofx;replacingittherefore byXwehave JD'"'{eX]=eai(D+a)~nX. Let-\/r(D)beexpandedinintegral powers positiveandnegative (ifnecessary)ofD;and leteatt>Xbeoperatedonbytheseintegral powersinsuccession, theequivalentvalues derived from thefore- going equations beingsubstituted andtheterms collected as before;thentheresult is Tjr(D)|e"X}=e"^ Corollary.Ifwewrite sothatYisafunction of#,then -. atheorem which isuseful. Forexample,letitberequiredtofind aparticularvalue ofytosatisfytheequation 44THEUNEAB EQUATION[33. But itunenotation adoptedthiswillbe iover .. orchoosingasothata+k=0,this is "D =-**IVe**dx. 34.Theorem III. If>/r(#a )beaneven function ofKthen fy(Da )sin(ax+a)=ty(aa )sin(ax+a). Tor Dssin(CM;+a)=(a")sin(a#+a), andthetheorem follows asbefore. Corollary.If^f(x)benotaneven function ofxitcanbe expressedintheform *(')+ X(^) where $andpareeven functions ofK;inthiscase ^(D)sin(oat+a)= {<j>(D1 )+D%(Ds j}sin(ace+a) =<(a*)sin(owe+a)+a^(a")cos(owet-a). Ifthefunction tobeoperated uponbethecosine instead ofthe sine, thecorresponding changesareobvious. 35.Theorem IV. This isreallyanextension ofLeibnitz's theorem forthesuccessive differentiation oftheproductoftwo quantitieswhose differential coefficients areknown. IfT/T(a;)asbefore denote anyalgebraicalrational function ex- pansibleinintegral powersofx,and^'(x\ ty"(x), -^r"(x\... denote its first, second, third, ...differential coefficients with regardtooc,then theextended theorem is uv Theproof dependsonLeibnitz's theorem and issimilar tothat ofthepreceding propositions. 35.] WITH CONSTANT COEFFICIENTS. 47 Theadvantageofthistheorem arises incaseswhere oneofthe twoquantitiesuandvisapowerofx,oristhesumofpowersofas. If,forinstance, u=a171"1 ,theseries ontheright-handsideneedonly bewritten asfarasthem"1term;andsuch inverseoperationsas aretobecarried outwillbeperformedonasingle quantityv. Ex.Shew that, if w* j"<".. (Z>+4)"--"7,W"'A' ^-\^f ''where 7isafunction ofxonly,yisgiven by --Cc /" [[ 36.Anotherimportant operator which sometimes occurs is as -j-or,with theprevious notation, xD;and similar theorems concerningthiscanbeenunciated. LetF(z) denote arationalalgebraicalfunction ofzexpansi- bleinpowersofz;then inF(xD) weshallhaveterms oftheform Jn J J (xD)nwhich means, not as"-j-^ ,butx-=- .x-*-...operatiug ntimes, Therelation between these willshortlybeproved. Theorem I. F(xtyxm=F(m)wm . For (xD)xm=ma?1 , ^t , */.' ri*' (xD}*xm=(a?D)mxm=m*xm , --^*''' ' ' and soforallintegral powers positive andnegative.Hence the theorem. Ex.Prove that ifZ7beafunction of#oftheform ''_ot-- -- ; then Theorem II.F(ccD)xmV=asmF(xD+m)V. Wehave scD(xm V}=xm (coD+m)V, or(a;""1.acD.oT) V=(xL+m}V, sothattheoperatorsaf .xD .xmandxD+mareequivalent. The course ofproofliesonlinesexactlysimilar tothose forthecorre- spondingtheorem withF(D) ;andtheresult isintheenunciated form. 48 THELINEAB EQUATION [37. 3"7.Therelation between theoperatorsD"andxDisgiven by theformula a?Dn=xD(xD-1)(tcD-2)...(.*Z)-+1). Thetheorem canbeestablisheddirectly;forifuthesubjectof operationbeexpandedinaseries ofterms oftheformAmxm ,the result ofoperatingonthiswithDnandmultiplying byosniszero ifm<n,and is m(m-1)(m-2)...(m-n+1)Amxm , ifm>n; butthis isalsotheresult ofoperatingwiththeright- hand side. Hence theoperatorsareequivalentforeachterm ofu andsoforthesum ofalltheterras ofu,i.e.foruitself. Thetheorem canalsobeestablished byinduction;forsuppose xnITu=xD(soD-l] (xD-2)...(asD-n +1)u, andwrite u=(xD ri)v; then Dnu=ccDn+1 v, andsoan+iDn+1v=xD(xD-1)(xD-2)...(xD-n)v. Nowuisanygeneral function; hence visalsoageneral function. Thetheorem,iftrue forn,isthus true forn+1;itis obviouslytrue forthevalues 1and2andsoistruegenerally. Some Properties oftheGeneral LinearDifferential Equation. 38.Thegeneral typeoflinear differentialequationofthe 71th order is inwhichXltX^,...,Xn,Varefunctions ofx(orconstants) but donotcontain y;forthesake ofshortness letitbewritten Ifthisequationbeintegrated stepbystepsothat each integrationreduces theorder oftheequation byunity, every time such areduction iseffected anarbitraryconstant enters, and therefore, whenultimately theintegral equationisob- tained,narbitraryconstants inallwillhave entered; orwe shallexpecttheprimitiveofagivenlinear differentialequation 38.] GENERAL PROPERTIES. 49" tocontain anumber ofarbitraryconstantsequaltotheorder of theequation. There arecertainproperties appertainingtoalllinearequa- tions incommon whichsimplifytosome extent theirintegration ; themostimportantofthese arethefollowing. 39. I.Letr)beanyparticularvalue ofy,which satisfies theequation ;and let y=i)+7. Thensubstitutingthisvalue ofyintheequation wehave OE>()7+<(.) 77=F. But, since17issome solution of theequation nowbecomes 3>OD)7=0, sothat tosolve theoriginal equation wemust solvegenerallythis equation, which isthesame astheoriginal equation exceptthat theright-handside isnow zero.When theprimitiveofthis modifiedequation,which willcontain narbitraryconstants because itisofthe 71th order, hasbeen obtained,itmust beadded to17;and theresult equatedtoywillbetheprimitiveofthegiven equation. Theprimitivethen consists oftwoparts: First, thequantity 97,which iscalled theParticular Integral and isanysolution whatever(thesimplerthebetter)oftheorigi- nalequation; Second, thequantity Y,which iscalled theComplementary Fu/nction;this istheprimitiveoftheequation when theright- hand side ismade zero. Thesum ofthese twopartsistheprimitiveofthegeneral equation.Ifinanyparticularcase theright-handsideshould alreadybezero,theformer ofthesepartswillnotoccur. The various methods available forthededuction ofthe Particular Integraloccur later in46.Theremaining properties areuseful intheinvestigationoftheComplementaryFunction. 40. II. IfF=7lbeasolution oftheequation F. 4 50 THELINEAB EQUATION. [40. thenY=C1Tlisalsoasolution, whereC^isaconstant; and if Ft,F,....... ,Fnbeparticular solutions, then isalsoasolution, where(7,,Ct,......,Cnareconstants. For 3>(D)Y=3>(D)C 1Yl+3>(D')CJ i+...... andeachterm ontheright-handside iszero.Norestriction whatever hasbeen laidonthevalues oftheconstants G,andthey therefore arecompletely arbitrary ;theabove value of7isthus theprimitiveoftheequation and soisthecomplementaryfunction intheintegralofthe equation Hence thedetermination ofthecomplementaryfunction is reduced tothat ofparticularsolutions ofthesubsidiary equation. 41.HI. Ifasingle particularsolution ofthesubsidiary equationbeknown, theorder ofthegivendifferentialequation canbelowered byunity. LetFxbeasolution of and letthesubstitution ofthevalueF^bemade intheequation 3>(D)y=V; then,by 35,theleft-hand sidebecomes inwhich theoperations on >-arederived from <3>bytemporarily considering Dasamagnitude andobtainingthepartialdif- ferential coefficients withregardtoD.But 41.] GENERAL PHOPEETIES. 51 andsoon;hence, re-writingtheequation, weobtain Butbyhypothesis <*>()*><), sothatthelasttermontheleft-hand side isremoved;thequantity Fjissupposed known andtherefore allthefunctions ofitonthe left-hand sidemaybeconsidered known. LetZbewritten for Dz;then theequation becomes anequationoforder n 1. Ex.Asacorollary prove that,ifmparticularsolutions ofthesubsidiary equation beknown, theorder oftheoriginal differential equation canbe reduced bym. 42. TV.Thegiven equation maybetransformed intoan equation, fromwhich thesecond term(Le.theterminvolvingthe differential coefficient oforder one lessthan theorder ofthe equation)isabsent. Thesubstitution ofY^zforygivesforthecoefficient oflF~*z Z^+nDY,, (anduptothispointinthelastsection theassumed value of7t wasnotused, sothattheequationthere wasperfectly general); since theterm inDn~^zistobeabsent wehave andtherefore v-aor rx=e, noarbitraryconstantbeinginserted asthedifferentialequation remains linear andofthew"1order. Ifthisvalue ofF,besubsti- i tuted, thedifferentialequationinzisfreed from theterm inZ)""1^. OfthesepropertiesI.and IE.willbeimmediatelyuseful. 42 52COMPLEMENTARY FUNCTION OPTHELINEAR EQUATION [43. General Linear Equationwith ConstantCoefficients. 43. Ifinthegenerallinearequationthecoefficients ofyand ofitsdifferential coefficients beconstants, itmaybewritten orsay f(D)y=V, inwhich/(Z))isarationalalgebraical integralfunction ofDalone, andVisanyfunction of cc.Ithasalready beenprovedthatthe solution oftheequationconsists oftwopartswhich canbeobtained separately ;these willbetaken inturn. 44.TofindtheComplementaryFunction. Thecomplementaryfunction istheprimitiveof Now ithasbeenprovedthat sothaty=eamwillbeaparticularsolution oftheequation,ifabe such astomake Butf(z)isarational, algebraical andintegralfunction of degree n,andtherefore there arenroots oftheequation /(*)=0. Letthese nroots bea,/S,...,X;then e*,ePx ,....&*are ri particularsolutions oftheequation andtheprimitiveistherefore y=Ae+6? inwhich A,B,...,Larenarbitrary constants. This value ofyis thecomplementaryfunction oftheoriginal equation ;and, ifthe rootsbeallrealanddifferent fromoneanother, itiscomplete. Ifhowever tworoots beequaltooneanother, saya.and&then thevalue ofybecomes y=(A+B)e** 44.] WITH CONSTANT COEFFICIENTS. 53 A^beingasingle arbitraryconstant(equaltothesum oftwo arbitrary constants). There arenowonlyn1arbitraryconstants iny,andtheexpression therefore ceases tobetheprimitive.In order toobtain theprimitive wemaysupposethat theroots are notequal but differbysomequantity hwhich willultimatelybe made zero;thepartdependingontheroots aandftwillthenbe AeP*+Be^+V* Asthequantities AandBarearbitrary, wemayassume them infinite insuch away that, ashapproaches zero,Bh isfinite andequaltoBI}whileAandBareofopposite signand their numerical difference(oralgebraical sum)isfinite andequaltoA1; thusthesumofthetwotermsAe*+BePxbecomes 3 ultimately, when hismade zero. Similarlyif '/roots beequalthecorrespondingrterms inthe complementaryfunction willapparentlycoalesce into asingle term;but itiseasytoshew, byreasoningsimilar tothatadopted forthecase oftwoequal roots, thattherterms willbereplaced a.denotingthecommon value oftherequalroots;andthecom- plementaryfunction willthenbe Ifnow theroots benotallreal, those which areimaginary must occur inpairs ;letsuch apairbe6+<jri*. The corre- spondingterms ofthecomplementaryfunction willbe which itissometimesnecessarytoexpressinaform freefrom *Thronghout thebookV^Twillbereplaced byi. 54COMPLEMENTAET FUNCTION OFTHELINEAR EQUATION [44. imaginary quantities.Ifcosine and sinevalues besubstituted fortheexponentials,thisexpression wiHbecome e6' \(A'+B)cos<f>w+i(A'-B')sinfa}. Since A'andB1arearbitrary constants, wemaywrite andwethenhaveFandGarbitrary ;thecorrespondingterms in thecomplementaryfunction therefore become eej>(Fcosfa4-&sinfa). Lastly, ifanimaginaryroot berepeated,theconjugate imaginaryroot will also berepeated and thecorresponding terms inywillbe Usingthesamemethod asbefore andwriting i(A'-ff)=Q, i(A"-B')=&', weobtain asthecorresponding partofthecomplementary function ee*{(F+F'x)cosfa+(G-+Q'sc)sinfa}. Resultsanalogoustothose inthecase ofmultiple repetition ofreal roots areobtained inthecase ofmultiple repetitionof imaginaryroots. 46.Insome cases ofthegenerallinearequation, when the coefficients arenotconstants butaresome functions ofto,amethod somewhat similar tothis willapply. Thus, itmight happen that, when foryintheequation there issubstitutedty(m,x),wherei|risafunction ofdefinite form, theresulting equation hadafactorindependentof#such as(ra) ;ifthjswereso,thefactor wouldusuallybeofthedegree n,and soequatedtozerowouldsatisfythedifferentialequation andwould furnish nvalues ofmwhichmaybedenotedbym^, 7a,...,mn;theprimitive would thenbe y=A&K,as)+A^r(ma,a)+...+An^r(mn,0). 45.] WITH CONSTANT COEFFICIENTS. 55 Iftworootswereequal,asm1andwa,thenwritingmB=m,+h wehave forthecorresponding partofy or onchangingtheconstants anmakinghultimatelyzero asbefore. Asimilarprocess holds forthecase ofamultiple repetitionof arootm l;andinthecase ofimaginaryroots thecorresponding partsofyshouldusuallyhave theconstantschangedinthe modifiedexpression,soastoleave thelatter freefromimaginary symbols. Thisprocess wasadoptedinthecase ofconstant coefficients, thespecial form of^usedbeinge;when theequationis homogeneous (55),that is,when ittakes theform inwhich thequantities Aareconstants, theproperform of ty(see 36)tobesubstituted isxn .Occasionally byasuitable changeofvariable agiven equationcanbereduced totheabove shape. Ex. 1.Solveg4 When wesubstitute y=dna ,theequationformis sothat y=(16-*+ Be-**. TO 7ffaAj fit! Ex.2. Solveg_2X Theequationformis sothaty=*Ccos(px+a), or e^"(Acosfix+BsinJLUC). Cor. Thesolution of is A*. Solve 56 THELINEAB EQUATION. [45. Theequation formis (m-1)2=0, andthereforey=&(A+Bx}. fSx. 4.Solve Theequation formis andthevalue ofyis (A+Bx)cosnx+(C+Dx)sinnx. When wesubstitute tf*fory,theequationformis m(m l)+wz.-l=0, sothatm=+1or-1andthevalue ofyistherefore -. 3S Ex.Q. Solve s3_3.^ cfco3cc c With thesame substitution asinEx.5,theequationformis or m3-6ma+12flt-8=0, givingm=2 thrica Hence thevalue ofyis 3 mbeing putequalto2after differentiation;andthustheintegralis a{-A+Slog+C(log#)2 }. &. 7.Solve Leta+6a;=2; theequationwillthenbesunilar inform tothelasttwo. Ex. 8.Solve (i) (ii) (iii) THEPARTICULAR INTEGRAL. 57 46.Returning now tothe linearequation,inwhich the coefficients ofthedifferential coefficients ofyareconstants,itis necessarytofindaParticularIntegraloftheequation inwhichFisafunction of ac.Solving bythemethod ofsym- bolicaloperators, wehave theevaluation oftheright-handside willfurnish asatisfactory value ofy. Insomeparticularcases theform ofFrenders evaluationeasy; wewillproceedtomention some ofthese which occur most fre- quently. I.LetFbearational, algebraical, integralfunction ofCD; supposethehighest powerofa;inFtobethe 71th .Tofindthe particular integral,..~.mustbeexpandedinascending powersof D;and,because Z)n+1andoperatorsofahigherorder would reduce tozero alltheterms ofF,theterms inthisexpansion beyond D" maybeomitted. Further, ifthelowest powerofDinf(D) beD*then theexpansionwillbeginwith D~*and itdoes not need tobecarried onbeyondDn ,i.e. ZT"**1**"1 ;hence inf(D]all terms oforderhigherthanDn+*mayinthis case atonce be omitted beforeexpansion. Ex. 1.Solve andthecomplementaryfunction ise2"(A+Bse) ;hence theprimitiveis y=e>* (A+Bx) Ex. 2.Solve(Ifi'-a4 )y= Theprimitiveisevidently 58 PARTICULAR INTEGRAL OFTHELINEAR EQUATION [46. Ex. 3.Solve(Z)4-2Da+D3 )y=a*. =p(1+W+ZIP+4JD3+52*+6-06 )j?, terms uptothe fifthbeing retained(46).Now1+2.D+... and^may beconaideredseparate operators; operatingwiththeformer firstandremem- beringthaionlyaparticular value iswanted sothat constants need notbe inserted with=3,thevalue foryis j"6 yA_+_ Now if jj2hadoperatedfirst(orifthesecondoperator hadbeentakeu dis- tributively, eachterm with^ ,soastobe tf tf*thenthevalue forywould havebecome Theprimitiveis andtheapparently additionalpartoftheparticular integral obtained, when theoperators aretaken inthesecond method,isseen tobeincluded in thecomplementary function, sinceCandDarearbitrary constants. Itiseasytoseethat ingeneral notmerelytheterms ofanorderhigher than _Dn+*maybeatonceremoved from/(.D), butintheexpansionitself all terms ofanorderhigher thanDnmaybeneglected whether thesubsequent operator D~*beofanordergreater orlessthan n.Inparticular,ifXbea constant, onlythelowest power needberetained. Ex. 4.Solve (ii) H.Thisnjethod maybeappliedtoevaluatey,whenVisan exponential, andtosimplifytheprocess (andsorender theevalua- tionmoreproximate) whenVcontains anexponentialfactor. In either casewemaywrite V=eaaX, 46.] WITH CONSTANT COEFFICIENTS. 59 andthen IfXbeaconstant, thevalue ofyisnowatonceobtainable bythe precedingmethod. Thequantity amayormaynotbearoot of f(z)0.Supposeittobearoot i1tunesrepeated,sothat for asinglerootr=1.Ifabenotaroot, r=0.Then expanding f(D+a)wehave /(D+<)-/ (a)+r-r" (a)+-., inwhichfw (a)means the/ithdifferential coefficient off(z)with respectto#,when aissubstituted forz;then forywehave(by attendingtotheremark attheendofEx.3onthelastpage) H Inparticular,ifr=0,then Ex. I.Solve Here 2isnotarootof+z+1=0, andtherefore andtheprimitiveis y=e~*a>(Aoos-^+Ssin-jr-J Ex. 2.Solve(JD-4Z)+3)y=2fl!)J =. 1 Here ym 1 2 andtheprimitiveis PARTICULAR INTEGRAL OFTHELINEAR EQUATION [46. Ex. 3.Solve(i)(Z)-a)n#=eM ; (ii)(*-6D+8)y=<P+(P*. Ex. 4.The roots oftheequation f(e)=0axeninnumber, being &!,Oj, ...,^ ;obtain theparticular integraloftheequation Discuss thecasewhentwooftheroots(a^andOj)areequal IfXbearationalalgebraical integralfunction ofxand thereforeexpansibleinpowersofas,then thequantity must beevaluated asbefore inI. Ex. 1.Solve (2 Here (+2)'" =< andtheprimitiveis ^ar. 2.Solve Here andtheprimitiveis * Ex. 3.Solve(i) (ii) III.SupposethatVcontains asineoracosine asafactor, so that *V'=Xcos(nee+a), inwhich nandaare.constants. Thenwehave toevaluate 46.] WITH CONSTANT COEFFICIENTS. 61 Let y1=:Xsin(nx+a), then Itnowremains toevaluatew 1 whichmaycome under oneorother ofthegiven rules;ifits value beu+iv,thenequatingrealandimaginary partswehave y=ucos(nas+a)vsin(nx-fa). InthecasewhenXisaconstant andcosneeisnotpartofthe complementary function, sothat inisnotarootoff(z)=Q)the evaluation isimmediate;forthen _ "/() Ifhowever COSTWJ beapartofthecomplementary function, sothatinisaroot r1timesrepeated, then since wehave wemustseparate andequatetherealandimaginary partsas before. Ex. I.Solve Then y= =realpartof a;GOBax 62 PARTICULAR INTEGRAL OFTBDELINEAR EQUATION [46. Ex. 2.Solve Then3 =realpartof andtheprimitiveis y=Acos#+.3 sin#+&#sinx. Ex. 3.Solve <(Z>)y=COBnx, ooa ?ia?notbeing aportofthecomplementaryfunction, Let Ifhowever cosn# beapartofthecomplementary function, then the denominator willvanish andapparentlyrender theparticular integralinfinite. But itismerely apartofthecomplementary function, multiplied byanin- finiteconstant, which may beabsorbed into thearbitrary constant; to evaluate theparticular integralitwould besufficient toevaluate assigning theinfinitepart(when hismadezero) tothecomplementaryfunc- tionandretaining thefinitepartastheparticular integral.Itishowever better insuch cases tousetheformer method;infact, thismethod isprefer- ableonlyinthecaseofexampleslikethatjust treated. Ex. 4.Solve" (i)jp+y=sm"#(bothwhennis,andwhen itisnot,unity) ; 46.] WITH CONSTANT COEFFICIENTS. 63 (iv) -jj|+2-&+y=$coscu;(when ais,andwhen itisnot,unity); (v) (vi) (vii) (JZ)2-2.D+4)2y=x(Poos (x) -j (zi) (xii)^+y=si IV. IfVcontain apowerofxasafactor, sothatwemay write then forthedetermination oftheparticular integral wemayuse theextended form(35)ofLeibnitz's theorem. Thus where theseries must becarried tothe(ra+1)*11term;each of these terms stillleaves aquantitytobeevaluated which maybe donebythemethods ofoneoftheprecedingdivisions;ifitmay not,thequantity maybeobtained bythenextmethod, which isof universalapplication. Thesuccess ofthisgeneralmethod depends solelyonthesolution ofanequation (thesolutionbeing requisite toobtain thecomplementary function) andontheintegrationof resulting expressions. 64 PARTICULAR INTEGRAL OFTHELINEAR EQUATION [46. V.Supposethat allthefactors, which occur inVandcanbe dealt withbyoneorother oftheforegoing methods, havebeen taken outside theoperator andthatthequantity remaining comes under none ofthese heads, sothatwehave toevaluate ex- pressionsoftheform Let-prbeexpressedinpartial fractions, eachhavingforits denominator alinear factor orapowerofalinear factor ofty(D}, theconstantquantities occurringnotbeing necessarilyreal;then thefractions willbeoftheform *n (D-a)n' where nisaninteger, Anandaconstants, and aaroot of ty(#)=0.Hence J- T-r- -* J-L- f~r- A =2A^I!......e-UdaT. Ifimaginary quantities enter intoanyexpression theconjugate imaginary quantitieswillenter intosome other;suchapairofex- pressions must ingeneral becombined soastoleave noimaginary quantityintheexplicit expressionoftheparticular integral. Ex. 1. Hence theparticular integralis 1)^3log*~ 2T^2logX=^ \*~*"log***~ andthecomplementary function is Ess. 2.Lettheright-hand sideinthepreceding example be a?loga;in- stead oflogs ;thenwemay eitherintegrate bypartsorusetheextension of Leibnitz's theorem. Thelattergives 4>6.] WITH CONSTANT COEFFICIENTS. 65 -eP l Ex. 3.Solve whereUisafunction ofx.Wehave ___ __ Zin\D-in D+in or,changing thevariable under thesignoftheintegral, \fx= -JUfBinnte-fidi-, inwhich Gisthesame function ofas7isofx. There isanother method ofintegratingthisequation which proceeds on different lines. Multiply throughout bysinnx :then d_fr dai\dx' andtherefore;j=U's.innx, -r-BiRTUB nycosnx=An+IC^si Similarly, multiplying byCOSTU; andwriting theequationinthecorre- sponding form,wefindanintegral -4-cosnx+nysinnx=Sn+ IU*cosnd. Eliminating-jrbetweenthese,weobtain y=Acosu?+JSsma;+- IU>sinn(x agreeingwiththeformer result. Ex. 4.Solve(i) +11*$=a?cosox, whenn> ctandwhenn=a; F. 66 PARTICULAR INTEGRAL OF[46. (U)g--v-ir, whereUiaanyfunction ofx; (Hi)g-%-4***. .Sir. 6.Bymeans of(ill)inJSfe.4provethat VSe"^2/"' xabf-'^dx-W^F tf^^dx^. ~V2' va 47.Owingtotheclosesimilaritybetween theLinearequation with constant coefficients andthehomogeneouslinearequation, thelattermaybedealt withhere;itmaybewritten intheform whereVisafunction ofxalone andmaybeaconstant C.Inthe latter case theparticular integralisatonce obtainable;itis evidently Iftheoperator x-5-bedenoted by S-,then(37) andthedifferentialequation maybewritten Consider thetwopartsoftheprimitive separately;thecom- plementaryfunction istheprimitiveof Nowwehavealreadyseenthat F(tya? Hence,ifpbesochosen that then ospisasolution oftheequation; and ifpltp^, ...,pB'bethe roots ofF(z)=0,thecomplementaryfunction is 47.] THEHOMOGENEOUS LINEAR EQUATION. 67 The case ofequalroots hasbeen discussedalready (45) ;if-7tworootsbeimaginary, sayp1andps,sothat/,/^j'.*/*'"";' r"''" p^=a+1/3andpt=a-)-"Vp''-' J_/^ then thecorresponding partofywillbe-^yc '-*'"'" 0*{^L;cos(0log*)+A;sinoslo,. t thearbitrary constantshavingbeenchanged.' <v- .,-- Ex. Iftheimaginttryroots az)8berepeated rtimes, thecorresponding part ofthecomplementary function willbe ,t,'' 48.Theparticular integralisthevalue of andtheevaluation maybeeffected intwoways, which arereally equivalent save forthedifference inoperators employed. IfVeither beapowerorcontain asafactor apowerofx, sayssm ,then ~W -r-u/fv \J- InthecasewhenTisaconstant, theevaluation iseasy.Ifm benotaroot ofF(z)=Q,thenwemayexpand {.F(Sr+w)}""1in ascending powersofS-andneglectallbutthe firstterm, which is independentofS-andinfactgives Thesamemethod(ofexpansion)willapplywhenTisarational integral algebraicalfunction oflogas;andsince S-logx=1, theexpansiondoesnotneed tobecarriedbeyond $",where nis theindex ofthehighest poweroflogIDinT.* Ifhowevermbearootr1timesrepeatedinF(z)=0,then 52 68 PARTICULAR INTEGRAL OF andwehave toevaluate[48. IfTbeaconstant 0,then since--r . . j , jc.1 .'*<"' i thevalue ofyis F"(m)' ifitbeafunction ofloga;asbefore, theoperatorshould beex- pandedinascending powersofS-upto" (S-r beingretained inthe denominator), andthevalue ofywillbegivenasthesum ofa number ofterms oftheform thatis,ofanumber ofterms oftheform .N (s+r) Ageneral expressioncanbegivenfortheparticular integralin thecasewhenVtakesnone ofthese forms. Let^-7^^eexpanded \ inpartialfractions andsuppose someterm tobe JLI/J^"^ A* thenywillbethesumofterms oftheform ^Ya_ ..K>;.&JJ-^ which isequivalentto "-Astf-^Var*, orAaf- 1Fa;-"-1da.* A J Another method ofproceeding ^istochangetheindependentj variable fromatoz,where tcis&;thischangesS-into -=-orD,dz 48.] THEHOMOGENEOUS LINEAB EQUATION. 69 and allthemethods of46willnowapply.Itiseasytoseethat allthecases indicated forS-arestrictanalogues ofcases indicated forD. Ex. Solve (vii).r2__ MISCELLANEOUS EXAMPLES. 1.Ifthere betwo linear equationsofordersmandn(?i>m)satisfied by thesame dependent variable, athird linearequationoforder nmcan without anyintegration bederived from the firsttwo;andtheequationsof ordersmandnm(when integrated)willsuffice tofurnish theintegralofthe equationoforder n.(Liouville.) 2.Solve theequations do? a:dx(a)ft+*-,v' <Py+ (y) 3.Prove thatthesolution of is= cos aj;-narccot 70MISCELLANEOUS EXAMPLES OFTHELINEAR EQUATION 4.Obtain thegeneralsolution oftheequation intheform y=f-*Kt(Acoarit+Bainrit) +- t[*c-i-*>Hin n'(t-f) ff'df,nJo where U'isthesame function oft1asUisoft,andn'isgivenby *=7l-ilA 5.Solve theequations ,-x&y (m)-A 6.Obtain thecomplementaryfunction oftheequation intheform andshew thatthepartoftheparticular integral correspondingtothetypical terms under thesummationsignis 7.Prove thatthesolution ofthe-equation s+e WITH CONSTANT COEFFICIENTS. 71 8.Prove that 9.Prove that (i)2an+ (ii)J)ms (iii) 10.Prove that,if3-denote x whereAQ,Alt...,4n_1arearbitraryconstants. 11. IfP,Q,Rbecommutative symbolsofoperationthesolution of ^av-ni,\* 2. .-i>"* j.v.C'j. -(sP'"'*' o^"* "^T'v't ' '.',"{' i r '%tu"'il Qfc*,*- CHAPTER IV. MISCELLANEOUS METHODS. 49.BEFORE wediscuss thelinearequationofthesecond order with variable coefficients there areseveral miscellaneous methods which itisadvisable toconsider;theyapplytosystemsofequa- tionswhich admit either ofcompletesolution orofapproachtoa solution intheshapeofafirstintegral.Itistobeunderstood that theequationshereaftergivenaretypical andnotmerely isolatedequationswhich canbeintegrated;itisfrequently possible toinclude others under some oneofthefollowingclassesbymeans ofwell-selected substitutions foreither thedependentorthe independentvariable. Such substitutionspointouthowever the limits within which themethods areforthemostpart effective, sothat itmust beborne inmind that themethods arenotof general applicationtoalllinearequationsofthesecond order. 50.Thesimplestcase ofallisthat inwhich theequation isoftheform whereXisafunction of asalone. Itisimmediately integrable andtheresult ofintegrationis A1denotinganarbitraryconstant. Asecondintegration gives 50.] MISCELLANEOUS METHODS. 73 Aybeing anotherarbitraryconstant.Proceedinginthiswaywe shallhave afternintegrationsasthegeneralsolution 2/= AinwhichBreplaces-.- r .and istherefore anarbitrary con- (n r)\J stant. 51.Anotherverysimple equationtobeconsidered is da?' inwhichYisafunction ofyalone;butingeneralitisintegrable onlywhen niseither 1,or2.Inthecasewhen nis2,letthe equation bemultiplied by2 :theneach sidemaybeintegrated, andwehave suppose;inthisthevariables canbeseparated andfinallythe generalsolution oftheequation Ex, Solve Afiratintegralis where oisanarbitraryconstant;separationofthevariables gives yandtherefore arcsin=ax+a,o or y=csw(ax+ a). 74 MISCELLANEOUS METHODS.[52. 52.Anydifferentialequationwhichmerely expresses arela- tionbetween two differential coefficients, whose orders differby either 1or2,admits ofsolution. Asatypeofthedifferential equation, when theorders differby1,wemaywrite d"y ,-, . 2. /r dxn dn~lv jn-i=-I7 ";th611theequation becomes theintegralofwhich is dY=0!+A. Supposethisequationcanbesolved forFandthatthesolu- tion is that is Then this isoneofthecasesalreadydiscussed(50),andthe general integralcanbeobtained. Orafterobtainingtheequation i|r(F)=x+A,wemayproceed thus :since therefore Similarly dY7d7_fdYf'hmi F(T)F(Y)} andsoon,until [jr_r*r_fYdY y-]F<Y)]F(Y) J, 52.] MISCELLANEOUS METHODS. 75 anarbitraryconstant beingintroduced after everyoneoftheinte- grations, which must betaken inorder fromrighttoleft.Then wehave twoequations between ao,y,F,fromwhichTistobe eliminated;andtheeliminant willbetheprimitive. Itisevident thatbythismethod theequation canbesolved. Esc. 1.Solve a-^= ,. cLc3dor then o-f ofwhich theintegralis K=M ^ andtherefore y=Aea+S.c+C, where A,B,Garearbitraryconstants. Ex. 2.Integrate 53.Asatypeofthedifferential equationswhich connect differential coefficients, whose orders differ by2,wemaywrite ~y.=z\then theequationbecomes dx*~* dz/./\ 76 MISCELLANEOUS METHODS. [53. thesolution ofwhich hasbeen obtained intheform dz /, If,after theintegrationshave been carried out,theEquation canbealgebraicallysolved forzinterms ofa,say z=B(as) (wherethefunction 9(as)will involve theconstants Aand5), then Ti2directintegrationswillfurnish theprimitive. But ifitshould beimpossibletoeffect thisalgebraical resolution, thenwehave TTd"~Hence- dz r zdz cT*y_f dz r <M"~J{A+2lf(z)dz\V{A+2lf(z)dz\V [A andsoon ;ultimately weshall obtainyasafunction ofz,andthe primitivewillbetheeliminant withregardtozoftheequation between yandzandtheequation between xand z.' *L Solve *g-g.I 4 When wewrite 2for-=-*[tlieecLuation becomea t i i^ I=?> sothat = flifl"+c2ea , andtherefore y=Aea+Bea+Cj;+I>, inwhiohAarid5replace Cja2and c2a2respectively. &r. 2.Solva 54.] MISCELLANEOUS METHODS. 77 54.Insomeparticularcases thegeneraldifferentialequation ofthesecond order can,bysubstitution, bedepressedsoasto become adifferentialequationofthe first order; such cases occur when oneofthevariables isexplicitlyabsent from the equation. First, consider anequationinwhich xdoes notoccur, sothat itmaybewritten intheform dx' da?) Let-f-=p andthen -r4=P-rStheequation thusbecomesdxrda? dy^ adifferentialequationofthe first order tofindpinterms ofy. Letthesolution be P= inwhich/(y)will include anarbitraryconstant. Then the variables areseparable,sincewemaywrite andintegrationofthisequationwilllead totheprimitive. Next, consider anequationinwhich ydoesnotoccur, sothat itmaybewritten intheform Let -r-P;then -j-^=-:theequation istransformed into dec^da? decn anequationofthe firstorder tofindpinterms of#.Letthe solution be p=F(x-), whereFincludes anarbitraryconstant.Integrating this,we obtain astheprimitive 78 . MISCELLANEOUS METHODS.[54. fi,.l.8*. te--I When we-write-/-=ptheequationistransformed into CttX __ 3a-y' theintegralofwhich is where/uisanarbitraryconstant;theprimitiveisgiven bytheevaluation of J &1 So. Thesubstitution -/=ptransforms theequationinto a2dp _J^-O/v. .J~^AiOi ^ onintegrationthingives andtherefore .sothattheprimitiveis^ y=\pdx=B+3* Ex. 3.Integrate 54.] HOMOGENEOUS EQUATIONS. 79 (viii) Homogeneous Equations. 65.There arecertain classes ofdifferentialequationsin which akind ofhomogeneitysubsists;andthesolution ofthese canbysuitable transformations bemade todepend uponthat of equations oflower orders. Thehomogeneityisconstituted as follows :ifybeconsidered tobeofndimensions, while#isofone dimension, then-/- ,since itisthelimit of~ ,isofn1dimen- cLx \a) sions;-^,beingthelimit of-^,isofn2dimensions, andso on ;andtheequationissaid tobehomogeneous when, ifthese dimensions beassignedtothecorresponding quantities, theterms are allofthesame dimensions. Thesimplestcaseis,ofcourse, thatinwhich nisunity. First, letnbeunitysothatxandymaybothbeconsidered of onedimension. Lety=xzandx=eP;then dy_fa $ dasdd ^ylf^zdz\ da*~(dP+ d6Je' andsoon;andtheresultingdifferentialequationwillbeonebe- tween zand 6.Now itwillbenoticed thatthecoefficient of6in theindex oftheexponential wherever itoccurs inanydifferential coefficient isthenumberrepresentingthedimensions ofthat dif- ferential coefficient;and therefore, when substitution takesplace inthedifferentialequation, supposed homogeneous,theindex of 6intheexponentialwillbethesame foreachterm oftheequa- tion,andthisexponentialwilltherefore beafactor whichmaybe removed. Thenewindependentvariable 9willnolongeroccur explicitlyintheequation,which will therefore beofthe class alreadydiscussed in54andcanhave itsorderdepressed. 80 HOMOGENEOUS EQUATIONS. [55. *. I.Solve * Makingthesubstitutions of55wehave d2*dz (dz N" * When wewrite^2=^1theequation becomes dv or,ifv=zs, andtherefore-r-=d6. Thevariables areseparated andtheequation canbeintegrated. Ex. 2.Solve Passing now tothegeneralcase inwhichhomogeneityiscon- stituted ontheassumptionofndimensions fory,wewrite Wenowhave andsoon. Itisobvious thatthecoefficient of intheindex of theexponential,which occurs intheexpressionofeverydifferential coefficient, ex.actlymeasures thedimensions ofthat differential coefficient; and asbefore, when substitution takesplace,the exponentialwilldisappear andthedifferentialequation, having been thus transformed into onefrom which theindependent variable isexplicitly absent, canhave itsorder lowered byunity. 55.] HOMOGENEOUS EQUATIONS. 81 Ex.l. Solve ^ This ishomogeneousifybeconsidered tobeoftwodimensions while a;is ofone. Hence wesubstitute andtheequation becomes S+' Afirstintegralisgivenby ---, andinthisthevariables canbeseparatedintheform theintegralofwhich willvary (beingeither aninverse circular function ora logarithm) accordingtothesignofA. Ex. 2.Solve '' /J~jj rf'K~~ \/7i / ff' w +-($)'- Aparticularsetofcases arises whennismadeinfinite;all thequantities y,-^-, have then thesame dimensions. The simplest method ofsolution istoadoptthesubstitution andtheresulting equationbetween uand aswillbeofanorder lowerbyunity than thegiven equation. Ex. 3.Solve F. 82 EXACT[56. ExactDifferential Equations. 56.Adifferentialequationoftheform -/d"vd^y dy \. /Off-^ *)-<> issaidtobeexact when, onrepresentingtheleft-hand memberby V,theexpression Vdw istheexact differential ofsome function U,which isnecessarilyoftheform Consider firstalinear exact differentialequation, whichmay berepresented by da?"-1 where thecoefficients are allfunctions of as.Anequationofthis form willnotingeneralbeanexact differentialequation, butwe proceedtoshew that,ifacertain relation besatisfied bythese quantities P,theequationcanbeintegratedonce. Indicatingforconvenience differentiation withregardtoasby means ofdashes, wehaveondirectintegration JPydtB= jPydas, ,jd*=-P,'"yd*+P fl" y-PBV andtherefore -P 1'+P1"-P8"'+......)ydx 8"-......)y G.! 56.] EQUATIONS. 83 where the-lawofformation ofthesuccessive coefficients Q,Qi}Qv... isthesame and,inparticular, Now thecondition ofintegrability evidentlyisthatthere shall benotermremainingwhich involves anintegralofy\andthe necessary andsufficientcondition forthis isthat <?=<>, thatis, When thjscondition issatisfied, the firstintegralis where A^isanarbitraryconstant. Ifnowthecoefficients Qsatisfythecorresponding condition,viz.:' theequationisagain integrable;andtheprocesscanbecontinued solongasthecoefficients ofeach successiveequationthus derived satisfythecondition ofintegrability. Ex. 1.Theequation isanexact equation ;forwehave P=l,P/=l, P2"=0,P8'"=0; andsothecondition issatisfied. Integratingeach sidewehave Inpracticeitissometimes easytoseethatagiven equationisinte- grable.Inmanycases thequantities Pareeither oftheform ozorsums of 62 84 EXACT[66. expressionsofthisform;and #"*-^isaperfectdifferentialcoefficient, if mbelessthann;forintegratingitbypartswehave Ifn=m+l thelastterm is(-\}mm\y. When weapplythislemma tothepresent example,thetermsinvolving^^lareseen tobeperfectdifferential coefficients, andx-+yis-j-(xy\dap da, o*a aa! sothattheleft-hand side isaperfectdifferential coefficient andtheequation istherefore exact. Ex. 2.Prove thattheequationinEx. 1cannot befurther integrated by theforegoingmethod, Ex. 3.Solve (ii) andshew thattheequation becomes integrableonbeing multiplied bysome powerof as.Obtain its integral. 57.Themethod which isused forintegratingexactequations which arenotlinearmaybeillustrated byconsideringanexample. Ex. 1.Solve Onthesuppositionthat this isanexact differential equation wemaywrite wherepstands for -r- .LetUdenote what would bethevalue ofUifp CvtC weretheonly variable,sothat Letallrestrictions beremoved,sothat dOl=(Zsp+2yp*)dx+(a?+2^) dp, andtherefore 57.] EQUATIONS. 85 whichgives onintegrationU-U thatis, andtherefore thefirstintegralis Theprecedingmethod willheseen tolead tothefollowing general rule fortheintegrationofanexact differential equationof the 71thorder. Theequation, beingderivable from oneoforder dnyn1bydirect differentiation, willcontain-3-^onlyinthe first degree ;ifthiscondition benotsatisfied, theequationisnotexact. Lettheequationbewritten intheformV=0,andintegrate Vdocasif *nj(were theonlyvariableoccurringinVand-j-^its differential coefficient;letthe result beU^Then VdasdUl involves differential coefficients ofyoftheorder nIatthe utmost; asitisanexact differential thehighestdifferential coefficient ofywhich occurs canenteronlyinthe firstdegree. Repeatingtheprocessasoften asnecessary, weshall ultimately have Vdx-dUl-dUt-...=Q. Then afirstintegralofthegiven equationis Ex. 2.Solve ...dydfy ,dywj-j-s.-y^j ^'dxdaradx Ex. 3.Shew thattheequation becomes Lategrable onmultiplication bythefactor 2#a-/-2#y.Hence deduce afirstintegral andtheprimitive. 86 LINEAR EQUATION OF[57. Ex. 4.Integratetheequation aV+ having giventhatthere isanintegratingfactor oftheformXj- (Euler.) Linear Equation oftlieSecond Order. 58.Weshall hereprovesome oftheleading propertiesofthe linear equationofthesecond order;butthepresent investigation willnotforthemostpart anticipatethediscussion ofthegeneral linearequation,forthepropertieshere established belong solely totheequationofthesecond order. Thegeneralform oftheequationis inwhich P,Q,andRarefunctions ofx;theymayinspecialcases bemerelyconstantquantities. Substitute intheequationforyavalue vw,where vandware both functions ofa;;asyettheonlylimitation onthem isthat their productmust beequaltoy.Wethenhave tfv Aswemaychoose arelation arbitrarilybetween vandwor make either ofthem satisfysome condition, wewillsupposeit possibletodetermine wsothat thecoefficient ofvmay vanish, thatis, da? doc which, itwillbenoticed,isthesame astheoriginal equationwith theright-handsideequatedtozero. Thequantity wbeingnow considered known, themodifiedequationbecomes . p= L das9\wdec Jdot)~w' 58.] THESECOND OBDER. 87 sothat %Le!Pd*=A+! cue Jw andtherefore f*?e-sr*+[^ Jw9Jvrdx. Ittherefore follows that, ifanysolution, whatever oftheoriginal equation with theright-handsideequatedtozerocanbefound,the complete primitive oftheoriginal equationinitsgeneral formcan alsobefound. Theproblemofdeducingthiscomplete primitive istherefore resolved intothat offindingsomesinglesolution of thesimpler equation. This, inthemostgeneralcase ofPandQ unrestricted toparticularfunctions ofa>,hasnotyetbeen effected; butinspecialinstances itispossibletodetermine such asolution asisdesired, sometimes byinspection,sometimes bymeans ofa converging series, sometimes bymeans ofadefinite integral;but inthetwo latter cases (whichareusually closely connected)the explicit evaluation oftheform obtained forvisdifficult orimpos- sible, thoughthisform(5)stillremains thesolution. Ex. 1.Solve Aparticularsolution of 3-- isevidently y=x ;hence writing y=xo intheoriginal equationweget Hence ^dx andtherefore Ifwi=0,thiscanbesimplified. 88 LINEAR EQUATION OF[68. Ex. 2.Solve (ii) 59. Ifhowever asolution oftheequation whenRhasbeen putzerocannot beobtained, then itissometimes useful toremove ~dv from thetransformed differential equationtheterminvolving-r-. CM? That thismaybethecasewmustsatisfy das fromwhich wefind w=e~Mpda> ; there isnonecessityforaddingaconstant intheintegrationasit willafterwards disappear.Insert thisvalue of^intheequation andwrite thentheequationbecomes Insomeparticularcases thisequationadmits ofimmediate solution, butthese cases occurmuch lessfrequentlythan those to which theprecedingmethodapplies ;andtheadvantageofthenew form,which willbeindicatedshortly,liesinanaltogetherdifferent direction. Nowweknow that ifasolution ofthisequationwith theright-handsideequatedtozerocanbeobtained, theprimitive ofthegeneral equationisobtainable;andwemaythereforequote theequationintheform Ex. 1.Solve, HenceP=--andtherefore 59.] THESECOND ORDER. 89 sothattheequation givingvis Thesolution ofthis is x andtherefore thegeneral integralofthe firafcequationis Ex. 2.Solve (ii)**_*.,.(,+ ),_x'efo2xdx\x*)* <ffi>-^ 60.Theadvantageofusingtheform instead of astypicalofthelinear differentialequationofthesecond order, lies inthefactthat forallsubstitutions such as/()foryinthelatter equation /isafunction ofPandQofsuchaform that,when the newequation has itssecond termremoved bythesubstitution Z=.iy ittakes theform Thus/isexactlythesame function ofP1andQlasitisofP andQ ;andwemaytherefore call/aninvariant ofthecoefficients 61.] OFLINEAR EQUATION. 91 then andtherefore #1 ^i y.^"*' sothat sisthequotientoftwodifferent solutions ofeither diffe- rentialequation. Wenow.proceedtofindtheequation which is satisfied bys ;jainceeach ofthequantities y(orv)mayconsist of twoterms eachcontaininganarbitraryconstant factor, thequotient ofonebytheothermaycontain threearbitraryconstants(not four, since withoutalteringthevalue orgeneralityofsuch a quotient anyconstant maybemadeunity);therefore the diffe- rentialequationsatisfied by s,afunctioninvolvingthreearbitrary constants, mustbeofthethird order. Indicatingdifferentiation withregardtocobydashes, wemay write v,"+Ivt=0. Taking logarithmsof *_^_^ s~ v, v.anddifferentiating it,wehave s_'= s which onbeingdifferentiatedgives s\sy But sothattheequationis 90 NORMAL FOEM[60. ofthe differentialequation*.Theequationsoreducedmaybe said tobeinits'normal form' ;andanytwolinearequations such astheequationsinyandzcanbetransformed intooneanother, if thenormal form ofeachbethesame. Ifitbeknown thattwogiven equationsaresotransformable andthe equationofsubstitution between thedependentvariables bedesired, thiscan easily beobtained byusing thenormal form asanintermediate transformed equation. Thus inthegeneral exampletheequationinybecomes transformed tothatinvbywriting andtheequationinvpassesintothat inzbywriting and therefore therelation which transformsdirectly they-equation intothe 2-equationis irz>/*, i =zey 'Ex. 1.Prove thattheequations and canbetransformed intooneanother; and findtherelation betweena, and x. Ex. 2.Find thevalue ofQwhich issuchthattheequation 3+-i+r- maybetransformed byasubstitution y=/(#)into Obtain thevalue off (us). 61.Letylandyabetwoparticular integralsoftheequation dor dxJ andv1andvathecorresponding particular integralsof d*v Of.Malet, Phil. Trans. (1882), p.751. 92 SGHWAEZIAN[61. andtherefore s or Differentiatingthisagain, wehave andthetranspositionofthelasttermgives This isthedifferentialequationsatisfiedbys;and itisofthe third order, aswasindicated. Thefunction ofthedifferential coefficients ofswithregardtox, which occurs ontheleft-hand side oftheequation, hasbeen called byCayleytheSchwarzian Derivative* and isdenoted byhimby {s,x] ;itissocalled because itspropertiesarediscussed and itis offundamentalimportanceinamemoirbySchwarz inCrelle's Journai(t.LXXV), thoughthefunction isnotoriginallydue to him-J-. 62. Ifnowanysolution ofthisequationcanbeobtained, then asolution oftheoriginaldifferentialequationcanbeimmediately deduced. Forletsuchasolution ofthenew*equationbedenoted bys;then since wehave, onintegrating this, .-Ob''*, whereGisarbitrary.This isonesolution;another is ^=vas=Oss, *Cayley, Oamb* Phil. Trans.(1880),vol. xiii.p.6. tItoccursimplicitlyinLagrange's memoir"Sur laconstrnotion deacartes geographiques" (Euvres, vol. iv.p.651(thisreference isduetoSchwarz), and Jaoobi'a Fundamenta Nova;andexplicitlyforthefirsttime inRummer's memoir onthehypergeometrio series inCrdle, t.xv.,which isreferred toinChaptervi.; seealsoCayley,I.c. 62.] DERIVATIVE. 93 andfrom these thecorrespondingsolutions oftheequation iny arederivedbyinsertingtheexponentialfactor. Whenanyone solution ofalinearequationofthesecond order isknown, wecan obtain thegeneralsolution;andhenceanyparticular value ofa satisfyingitsdifferentialequationwillleadtothecomplete solution ofthe firstofthedifferentialequations. Thistheorem holds inregardtothegenerallinearequation of thesecond order;but itschiefapplicationariseswhen thelinear equationisthat satisfiedbythehypergeometric series, tobe discussed inChaptervi. Ex. 1.Prove that, if s(flU?+&)=OF+fl?,. theSchwarzian derivative ofsvanishes. I/I/>'-Ex.1. Find thegeneral value ofswhenpi4/r-^/ x*{.x]+a=0,/ef*Ji<w where aisaconstant. Ex. 3.Prove that' ,., , i . r i , , (iv) {,,,}- {,,y]~K}+ (y,r}. 63.Another method which issometimes effective isthat of changingtheindependentvariable. Take zasthenewindependentvariable ;then ^ dxdzdss' fy_d*y/dz\* dyd*z m a?d\dos) dzdo?' andtheoriginal equation becomes *dy(d*z+jj- adz\dso 94 SOLUTION BYCHANGE OFTHE[63. Asyetzisquite arbitraryand itmaytherefore bechosen to satisfy anyassignablecondition. Thuswemaychoose tomake dtj thecoefficient of--vanish sothat dz ^ da? dx' andtherefore zisgiveninterms ofwbytheequation .z= Theeliminant 6fthisrelation between zandxandthetrans- formedequation mayfurnish adifferential equationwhichproves integrable. Oneintegrablecase occurs when thevalue ofzissuch asto satisfytherelation daoj i"Ucy" where\iisaconstant;andthen theequationtakes theform ofwhich theintegralis aand#beingtheroots of m(m 1)+fi= ; and itisnotdifficult toprovethat therelation which must exist between PandQinorder that thismaybethecase is Anotherintegrablecasewould befurnished by andsoforother cases;and itwillbenoticed thatineach casethe equationisreduced towhatmaybecalled aknown form, thatis, oneofwhich theprimitivecanbeobtained. 63.]INDEPENDENT VARIABLE. 95 Ex. 1.Solve a-*--* Here P(l-x*)=-x, ? fxdx sothat *-,-!**- Ji=* O.S -(!-*)-*, and e=arcsin JT. When theindependentvariable ischangedtoz,theequation becomes &y*-&'** andtherefore (i) 64.Thepropertyused in60toobtain therelations between thedependentvariables intwoequations, which aretransformable into oneanother viz.thattheequations have thesame normal form canbeused toobtain therelations between thedependent variables intwoequations,theindependentvariables inwhich are different, onthehypothesisthat theequations ultimately determine thesame function. Theprocess adoptedwillbesimilar totheformer one,asbothequationswillbereduced totheirnormal forms inthesame variable andthese, beingassumed identical, will givetheconditionsnecessaryforthejustificationof'thehypothesis. Letthetwoequations,which aretobethustransformable into oneanother bychangingboth thedependent andtheindependent variables, be EQUTVAIJENT [64. +Cy= .....................(i), inwhich.PandQarefunctions ofso,andRand5functions ofz. Writingin(i) "*-*. and /---*. wehave andwritingin(ii) and J=S--E', cZauwehave-j-+/Vj= ........................(iv).Ct* In(iii)changingtheindependentvariable fromxtoz,we obtain ' inwhich dashes indicate" differentiation withregardto ac.To reduce thistoitsnormal formwewrite or,ontheevaluation oftheintegralintheexponent, yX*=ys; theequationthenbecomes 64.] EQUATIONS. 97 where 7_i^'_1/*"'jM and{.e,a;}istheSchwarzian derivative ofz. If,then, theequations betransformable intooneanother, the normal forms willbethesamewhenexpressedinterms ofthe sameindependentvariable;hence comparing (iv)and(v),which arethenormal forms, wehave and G=J. SubstitutingforGinthelatterequation wehave or ''-+ andsubstitutingtheir values forytandv,intheformerequation wehave These twoequationsaretheconditions that thedifferential equations (i)and(ii)should have thegiven property.The firstof themgivestherelation which must existbetween theindependent variables; and,when the first issatisfied, thesecondgivesthe relation which must existbetween thedependentvariables. Theforegoing equationsenable ustoobtain thegeneralform ofalldifferential equationsintowhich(i)istransformable, and alsotoobtain theconnexion between twogiven related equations. Thus, forinstance, theequationinagiven independentvariable zequivalentto(i)would have asitsnormal form F. 98 EQUIVALENT EQUATIONS. [64. ,/dz\ rpdawhere i)=yI-y- J&, , ,I .[z, oc]and l's=P~"s i' and since and/areknown interms ofas,Jisalsoknown in terms ofa?andcantherefore beexpressedinterms ofe.Every differential equation,which isequivalentto(i)andhaszforits independent variable, must have theforegoing equationinv lfor itsnormal form. Ex. 1.Prove thattheequations j /,^l-3*dand (i aretransformable intooneanother bytherelation *(1-#)=!+*; andfindtherelation between zand v. (G.fl.Stuart.) JEx. 2.Prove thattheequations ,and js j- ds? xdxy aretransformable intooneanother bytherelation xl=xz; andfindtherelationbetween yand v. MethodofVariationofParameters. 65. Itwasproved (58)that ifasolution oftheequation -g+*!+*-o beknown, theprimitiveoftheequation 65.]VARIATION OFPARAMETERS. 99 canbeobtained; butthefollowing method iseffective ingiv- ingforthis(and other linearequations) what wascalled inthe lastchaptertheParticularIntegral,and itcanbeapplied where the methods formerly indicated cease tobeapplicable. Letylbeasolution oftheequation sothat .'+* Eliminating Qwehave y*3!-v*+p y'<Wyda?+^ andtherefore y*3L-Ayidxydas .i L-ff-s^JL ofwhich theintegralis '"7 Let?/astand forthequantityofwhichAisthe coefficient, so thattheprimitiveis andyaisaparticularsolution ofthedifferentialequation. Then thepreceding analysis shews thatanytwoparticularsolutions yt and2/aareconnected bytheequation where thevalue ofGisnolonger arbitrary butdependsonthe forms ofyLandy^,thetwoparticularsolutions oftheequation. t> 66.Letusnowtaketheabove value ofyandsubstitute itin theequation 72 100 VAEIATION OF[66. onthesuppositionthatAandBarenolongerconstants hut functions ofa;tobesochosen thattheequationshallbesatisfied. Thus theformofyisthesame forthetwoequations,butthe constants which occur intheformer casearechangedinthelatter into functions oftheindependent variable; tothisprocessis appliedthename Variation ofParameters. Wehavenowtwounknownquantities AandB,interms of whichy,asingle unknown, isexpressed ;andwearetherefore at libertytochoose anyrelation between them thatmaybemost convenient forourpurpose. When wedifferentiate yweobtain dy -ody, AdyadB dA .da das provided dB dA .^^ar-' weshall take this lastequationastherelation between AandB. j Again,ifwedifferentiate-^-,sothat &y=$d*y<i+^<*"& ,dBdvi \dA<jy. da? da? da? dxdm dasdx' andsubstitute these values intheoriginal equation, then, sincey^ and2/8areparticularsolutions oftheequation whenR=0,wehave astheresult dBdy^dAdyt_ docdec dxdx Thus dAdB_ dx dx_R Rjpdx ~y'is=~*~v<*y*-.v^~v 2/1day*dx andtherefore 66.] PARAMETERS. 101 whereEandFarearbitraryconstants andCisanabsolute con- stantdepending upontheforms ofylandyr Ifnow inthedifferentialequation wewrite$(x)forPand ty(a)forR;fj_(so)forytandft(x)forya;then thegeneralso- lutionof is y=tf/t()+IK()+* JV(0*Jf*(*)d* I/.(*)/i wherefi(co}andf 3(a)areparticularsolutions of arethereforeconnectedbytherelation Itmaybenoticed thatwemaymakeGunitywithout lossof generality ;forifitbenotunitywemaysubstitute forf^(as)the quantity-^f^(#)which, while stillaparticular solution, willrender theconstantunity. Ex. 1.Solve Arrangedintheordinary form this is d*y xdy ,1- Particular solutions oftheequation without theright-hand member arex and e*;hence,ifwetake A(*)-*i /(*)-" -wemayproceedasabove, andhave astheprimitive AsinthegeneralcaseAandBareconnected by' 102 VARIATION OF[66. ,., dA dBwhile~j~e^~l=x 1. d'A JD Thus^= andtherefore and B=F-x. Theprimitiveistherefore Ex. 2.Integrate bythismethod theequation whereQandRarefunctions ofxalone. Jfo. 3.Solve (i) 67.Themethod ofvariation ofparameters maybeapplied inamanner, different inregardtothetermsneglected,toobtain asubsidiary integral,theconstants iuwhich aresubsequently made variableparameters.Thus consider theequation Neglecttheterminvolving F(y]inorder toobtain asubsidiary integral;itwillbethat of dor which is- $L. da Suppose nowthatCinstead ofbeingaconstant isafunction ofxand letthisbedifferentiated;then da' 67.]PARAMETERS. 103 Or j;\iiio j.dx Therefore andso Afirstintegraloftheoriginal equationtherefore is |V/(^=[A- Thiscanbeagain integratedsince thevariables areseparable. Ex. 1.Solve inthismanner theequation Shew alsothattheintegralofthisequation maybederived bythemethod of 54.fcw^fl,f.|$I Bychangingtheindependent variable inthisexamplefromxtoytobtain theintegraloftheequation Ex. 2.Integrate thegeneral equation firstly, byneglecting thelastterm toobtain asubsidiary equation andthen varyingtheparameters ; secondly, byapplyingthesamemethod totheintegralderived fromneglecting thesecond term; thirdly, bymultiplying by(-j\andthen integratingeach term. Itthusappears from theseexamplesthat isintegrableinthecases : (a)when bothPandQarefunctions ofx, (]3)when bothPandQarefunctions ofy, (y)whenPisafunction ofxandQafunction ofy 104 SPECIAL METHODS FOR[68. Twoparticularmethods. 68.When intheequation -v-j+Iv=thequantity 7isa rationalalgebraicalfunction ofafractional form such that the denominator isofahigher degreeinthevariable than thenume- rator, thefollowingmethod issometimes ofuse. Letaquantityz/* besubstituted forv;then theequation becomes where Onintegratingtheequationasiftheleft-hand sidewere a perfect differential, wehave da: Since thequantities P landPaareconnected asyetbyonlya single relation, wemayassignasafurther condition todetermine them andthisgivesastheequationforPl *Lp.-r dx^-^ while,ifanyvalue ofPxsatisfyingthisbeobtained, anintegralof theoriginal equationisobtained intheshape Itshould bepointedoutthattheutilityofthismethoddepends ontheformoftheequation whichgivesP1;thiswould belostby thesubstitution p_1dw 1wdx' forthentheequation givingPtbecomeschangedtotheoriginal. 68.] LINEAR EQUATIONS. 105 'With theassumption which wasmade as-totheform ofIwe maywrite TVVUUV say,where T,Z7andVarerationalintegralandalgebraicalfunc- tions ofas.Thenwemayassume leavingtheconstants in/(#)asthequantities'tobedetermined from theequation; butingeneralthere arenot sufficient disposable constantsarisingin/toallow theequationtobesatisfied. Hence thismethod, liketheother methods which havebeenproposedfor thesolution ofthelinearequationofthesecond order,isnotone ofuniversalapplication, but iseffectiveonlyinparticularcases. Ex. 1.Solve x(1-x^-r^=2w. Here theequationforPis dP 2 j 1 /1 flj"\2" EFLetPx=+=- ;andsubstitute;theequationissatisfied byE=F= -1, andtherefore afirstintegralis 2_. , .v fdx fdxwherelog-=-I I, ,zjxjl-x or vx=z(\. -x}. Theprimitivecaneasilybededuced,fortheequationislinear ofthefirst order. Ex. 2.Solve (ii) 106 SPECIAL METHODS FOR[68. Ifaterminvolving -^-should occur intheequation,thisterm should beremoved beforeapplyingtheabove method. Ex. 3.Solve d*yy-(a+$)xdy ay__WdaP* *(!-*) dxx(\-x}~' a?Jy afly n. *(!-*) ?.4.Shew that thismethod willapplytotheequation da? provided there beasinglerelation between A'tB'andC";and find this relation. 69.-Acertain class oflinear differentialequations canbe solved bytheresolutionoftheoperatoronyinto theproduct of operators.Thus consider theequation inwhich u,vandwarefunctions oftc;then, iftheoperator ufL+vsL QjOj (LOG beresoluble intotheproduct d.Vd p,q,rand sbeingfunctions ofas,theequationcanbeintegrated. For, ifwewrite dz** wehave p+qz=0, cwc andtherefore =Ae~* Pdas , 69.] LINEAR EQUATIONS. 107 andwemustnowintegrate which islinear ofthe first order. Inorder that thisresolution maytakeplace,wehave thethreeequations todetermine fourquantities p,q,rand s;butwemayconsider pandrasknown factors ofuandtreat thetworemaining equa- tions todetermineqand s. Bnt these cannot hesolved ingeneral, andagaintherefore the method willapply onlyinparticularcases. Ex. l.Solve (x*+a;-2)-p(+(a?- a?)-^-(GaP+7#)y=0. Herewemaywritep=a+Z andr=x-l. Ifq=Ex+F ands=E'x+F', wehave '=-2 which aresatisfied by TT*^_o.TTTI_^_ga7^^ zL ff" 1 Hence theequation maybewritten Afirstintegralis andtheprimitiveis 108 PARTICULAR FORM OFEQUATION.[69. Ex. 2.Solve (ii)(*-l)(*-2)-(a.- (ill)^- (iv) (vi)^(a-fa)-2tf(2a- 70.There iaaparticularform intowhich theordinarylinear differentialequationofthesecond ordermaybechanged ;multi- plying throughout byff^J?d!B ,wemaywrite it Letanewindependentvariable zbetaken such that dz=Q</Pd"dK- then theequation becomes *W/M . rfz(/ dzj* Now Qe-^*" isadefinite function ofxandtherefore of.z; letitbedenotedby-~,whereUisafunction ofz.Then the equationis which istheform referred to. SirWilliam Thomson hasindicated amethod ofapproximating toasolution ofthisequation bymechanical means*. *SeeProa. Roy. Soc.Vol. xxrr.(1876), p.269. 70.] GENERAL LINEAR EQUATION. 109 Ex. Expresspfa&+Q.+Ru=Q"*theform foa+Pv=aQ -Prove v=S-Sl+S}-...,where S-C+C'x, Sn+l=I'da \pSndx,JoJo expresses thesolution ofthis inaseriesnecessarily convergingforallvalues ofx,provided p,remains finite. Work outthecasewhen /*=#". General Linear Equation. 71.Thegenerallinearequationwith variable coefficients isof theform inwhichX,X1}Xa,......,XnandVarefunctions ofacalone;the class inwhich the coefficient9 ofthe differential coefficients of yare"constants hasbeenalreadyconsidered. The coefficientsX,X1}......,Xnmaybetaken tobeintegralfunctions of#;if inanyequation theywere notactually so,theequation could be transformed sothat itscoefficients would beintegralfunctions of xbymultiplication throughout bytheleastcommonmultipleof thedenominators ofsuch fractions asoccurred inthegivenform. Theprimitiveofthe differentialequation consists, asbefore, oftwoparts: First. TheParticularIntegralwhich isanyvalue ofy(the simplerthebetter) satisfyingtheequation ; Second. TheComplementaryFunction which isthegeneral solution oftheequationwithout thesecond member, thatis,ofthe equation Theequation (ii),beingofthenthorder, willhave initsgeneral solution narbitraryconstants thenecessary number forthepri- mitive of(i),which isthesum ofthese twoparts. 110 GENERAL LINEAR[72. 72. Ifylbeasolution of(ii),thenA^isalsoasolution since theequationislinear;andtherefore, ifylfys,......,ynben differentparticular solutions of(ii), whereAI}A........,Anarearbitrary constants,isalsoasolution. Ifnowthesolutionsy1}ya,......,ynbeindependentofoneanother sothatnooneofthem canbeexpressed bymeans ofalinear function ofall,orofany of,theothers, then theforegoing value of yisasolutioninvolving?iarbitraryconstants;itistherefore the ComplementaryFunction. Inorder that thismaybethecase theremust benoequationoftheform foranyvalues whatever oftheconstants\,\,......,Xnother than zero foreach ofthem. Ifalltheconstants Xbenotzero,wehave thederivedequations \if1 i% \J2 l I\ Q /\ 7n-g+N 7n-^+ +^"n ?n-g="jcueBcte"dx and, since theVsdonot allvanish, thedeterminant obtained byeliminatingtheX'smust vanish, thatis, \ rii ::i ::za n cZ^1'da"-1''"'da"-1~ '"'da da' dx''"'dx Hence thecondition that they'eshould beindependent or, inother words, thattheforegoingvalue ofyshould betheCom- plementary Function, isthatAshould notvanish. '73.] EQUATION. Ill 73. Itiseasily proved that,ifAbezero,thensomeequation oftheform must exist. Forotherwise letthevalue oftheleft-hand sidebe denoted byu\multiplythecolumns inAby\,\.....,\respect- ivelyandaddthemtogether, replacing some onecolumn asthe first bytheirsum. Thenwehave doT1'dz"-1'do?-1 "da?-*''"'da=0, u, ya, ..., yn anequation^oforder n1which determines u.Now this issatis- fiedbyu=yltya,...,y n>thatis,ithasnparticularsolutions which aresupposed independent. Butthenumber ofindependent par- ticular solutions which anequationcanhave isequaltoitsorder, aproperty which isviolated bytheprecedingresult. The fore- going equationinumust therefore beanidentitysothatuiszero andtherefore, onthesuppositionthatAiszero, there isarelation between thenquantities y. 74.Thevalue ofAwhen different from zerocanbefound as follows. Letthevalues y=yi,y^,....ynbesubstituted in(ii)and from thenresulting equationsletthecoefficients J5Ta,Xa,...,X^ beeliminated;thenwehave dxn das'1 'daT"'"'drt da"-3' y.. 112.PARTICULAR INTEGRAL OF [74. Thedeterminant which ismultiplied byXis-j ,andtherefore thisequationis which whenintegrated gives A=Cff- SinceAand IX^'1da;aredeterminate functions ofas,the constant must bedetermined bysome other method;compari- sonofparticularterms isoften effective. Thevalue ofGwill evidently changewithachangeinthesetoffundamental solutions 2/1.2/9.-.2/n- Ex.Lety-ibeaparticularsolution oftheequation when wewritey-^adxfory,theequation determiningsis("76,post}oforder TTi 1.Letz-jbeaparticularsolution ofthis,sothaty-^z^dxisasecondparti- cular solution ofthe^-equation;and lets^udscbesubstituted for e.Thus theequationinuisoforderm 2.Leti^beaparticular solution ofthis equation ;theny^z^dx^dxisathirdparticular solution oftheoriginal equa- tion. Proceedinginthiswaybym 1successive substitutions weshall arrive atanequationoftheform dw. =-=tw,da;' ofwhich asolution canbefound;andthere willbe,inall,wtparticular solutionsy. Prove that theseparticular solutionsyareindependent ofoneanother; andshew that forthis setofparticular solutions (Fuchs.) 75.TheParticularIntegral maynowbededucedbymeans of themethod ofthevariation ofparameters ;this isthemostsym- metrical method, butanother willbeindicated inthenext section. Intheequation Scott'sDeterminants, p.36. 75.]THEGENERAL LINEAR EQUATION. 113 letthe.4'sbesupposedfunctions ofxinstead ofconstants;then thevalue of-isgiven by ......... docadas doc dA dA dA. Now aswehavenfunctions A,white theonlycondition asyet attached tothem isthattheyaresuch astomake thepreceding value ofysatisfythedifferentialequation (i),wemaymakethem satisfy n1other conditionsassignedatpleasure, providedthese arenotinconsistent. Let-^gassume asoneofthese conditions x dA. dA.'dAny^+y'~K+........ :+ s'--sr=0- andwethenhave Differentiatingthisagainwehave providedweassignasanother condition ...... == dxdx dx due........dxdx Proceedinginthiswayandassumingthatthe^.'Baresuch as tosatisfy d^dA dfytdA<PyndAn_ dx*dx^da?dx^.........*da?dx~ ' dx*dxdo? dx......... dx*dx rf^idA,,Q/2dA d^y ndA dxnuidx+ dx"-* dx+^dx"~* dx' (whichwith theprevioustwomake uptheassignablenIcon- ditions, notinconsistent) wehave F. 8 114 PARTICULAR INTEGRAL OF[75. The lastofthese,when differentiated, gives dec' but, asalltheconditions which wereassignable havebeen used,the secondpartoftheright-handsidedoesnotvanish. Ifwemultiply thedifferential coefficients ofythusexpressed bythealgebraical coefficients which areattached tothem intheequation (i)of71 andaddtheresults, sinceyisasolution of(i),andy^yv......,yn aresolutions of(ii)of71,weshallhave.-T1da rfaT1das.........c&T1das' dn~VLetA,betheminor of,._,'' inAforthevalues r=1,2,...,njCMC ., ., ,.j.v ifdA. dAa dAnthen thenequations givingthevalues ot T-*,-,-* ,.__........,7- have astheir solution theequation forallthevalues ofr.Hence dA. 7A. i andtherefore whereCrisanarbitraryconstant. Thevalue ofyistherefore r=n (ry\ ) -y=*y rk+/3^^, r=l(.J-^oaJ theParticularIntegral being FA 7.5.] THEGENERAL LINEAR EQUATION. 115 Ex. 1.Shewthat,if/L(x\/2(#),/g(as)bethreeparticular solutions of theequation inwhichQandSarefunctions ofxonly,thenthecomplete integralof d d isgiven by /i(ft/,(&MQ where1}Ct)Oaarearbitrary constants andaisadeterminate constant. Ex. 2.Solve theequations W^JLa~ : 76.When weknow oneorseveralparticularsolutions ofthe equation (ii)of71,theorder oftheequation canbedepressed bya numberequaltothenumber ofparticularsolutions known. Thus supposeweknow thaty,isaparticularsolution oftheequation; whenwechangethevariable fromytoy^utheequation becomes iV'^~rri iJ^+Ai +u/1^"^r or,what isthesamething, .p,ffu ,,,d11'1u-p.,duX^w+z*-d^++z-isr' inwhichXL',Xa', ,X'n_larefunctions ofXvXlt,Xn_1and differential coefficients ofy,.Ifnow for-?-wesubstitute v,the "dx resulting equationisoforder n1andtheoriginal equationhas therefore had itsorderdepressed byunity.82 116 DEPRESSION OFTHE[76. Ifytbeanotherparticularsolution of(ii),thenyjy 1isavalue ofu,andtherefore-j- ( )isasolution oftheequation inv;andCMC\yj* ' thiscantherefore have itsorderdepressed byunity andtheorder ofthenewequationwillbelessbytwothan that of(ii). Itwill beseen tobepossible byproceedinginthiswaytodiminish the order ofanequation bymwhenmparticularsolutions areknown. Each depressed equation remains linear. 77.When nIparticularsolutions ofanequation ofthe ?ith order areknown, theequationcanbedepressedsoastobealinear equationofthe first order, and asthelatter canbesolved, it follows thatwecanobtain theprimitive ofanequation ofthenth order whennlparticularsolutions areknown. Thefollowing method ofobtainingtheprimitiveavoids theprocess ofsucces- sivedepressionsofthedifferentialequation. Letthenlparticularsolutions oftheequation (ii)berepre- sented byyvya,y^;and letOvC3C^ben-l functions ofxsuchthat isasolution of(ii) ;asthis istheonlyrelation between thenl functions, wemayassignatpleasure n2other relations, provided theyarenotinconsistent. Letthese be dC dC dO. . , _dxdocdxdx...... docdx _ dx do?^ dx~' then thevalues ofthesuccessive differential coefficients ofyare given by 77.]ORDER OFTHEEQUATION. J-l, dn~l 1J d"~*V tZ""1 '!i*j___/^^"yii/nf jflI i/^ b117 --idOrrf*y'--1tfQ Thesubstitution ofthese values intheequation (ii)gives sinceyvy,.......... ,2/B_iareparticularsolutions. LetAdenote thedeterminant da?-*' 2/ 'V and lettheminor of-jrgginthisbedenoted byArforthevalues r=l}2,......... ,7i-l. Thenwehave f da' dx da andtherefore forthese values ofr Hence 118 DEPRESSION OFORDER OFEQUATION. [77. and "--^cry_r---i<n, r_ -!dadx-*r r*A-d^-*Ai Also ckc" dxTdas andrp_ ridadaT1 ' sothat '-^tfaQ^cfc'-S-icT3 .yr_ <fc. r^i^dor8iter=l^""8Cte' thetransformedequationtherefore is Dividing by-2"Awehave ds ,/2 j~+IAaa;\A theintegralofwhich is Thecorrespondingvalue ofCrisderivable from -,-&*-r*e "docandtherefore forthevalues r=1,2,.........,n 1.Hence wehavewarbitrary constants, viz. J.,J^,J.2,.........,An_i;andtheprimitiveof(ii) isthus r=n-l r=n-l rA -.f^idaj y=2^ r+ASyJ^aJ^dx. r=l r-1J^ r Ex. Solve completely wherePand areanyfunctions ofx. 78.] TRAJECTORIES. 119 GeometricalApplication: Trajectories. 78. Ithasalready been noticed thatadifferentialequationis theappropriate analytical expression ofanypropertyofacurve which isconnected -with itsdirection anditscurvature; and itthere- forefollows thattheinvestigationofmany geometrical questions ultimately depends uponthesolution ofadifferentialequation. Inthehigher partsofmathematics differentialequationsareof almost universal occurrence; butinothersubjectsitislesspos- siblethan itisingeometrytogiveexamples,asthere isnoneces- sarily generalmethod ofarrivingatthe differentialequation, while itsdeduction ingeometrical problemsisobtained almost immediately bytheuseoftheformulas ofthedifferential calculus. There willbenoattempttogivehereauycompleteclassification ofapplicationstogeometry;there willbeonlyasingle general problem discussed, thatofTrajectories. ATrajectoryisdefined tobealinewhich, atitspointsof intersection withthemembers ofafamilyofcurvesexpressed by oneequation,cutsthemaccordingtosomegivenlaw. 79.Asthemostgeneralformpossible,let f(x,y,a}=Q denote afamilyofcurves ofwhich aistheparameter ;through anypointononecurve atrajectorywillpassandthere willthus beasecond systemofcurvesrepresentingthesetrajectories.Let fand77bethecurrent coordinates 'ofthissecondsystem;and supposetheanalytical expressionofthelawwhich holds ateach pointofintersection tobe dy d?y dmd*7)*a? **al'w Inthisequationatapointofintersection and?;arerespectively thesame as asandy,beingthecoordinates oftkatpoint;but j arenotthesame as-^, ,fortheyindicate the a ax direction andthecurvature ofthetwointersectingcurves. 120 TRAJECTORIES.[79. Weproceedaafollows. From theequation f(x,y, a)= weobtain thevalues ofallthedifferential coefficients ofy,which occur intherelation F=0,asfunctions ofx,yanda;and in each oftheseexpressionswesubstitute thevalue ofaasafunction ofxandyderived from theequationofthecurve. This willbe equivalenttoeliminatingabetween /=andtheequation giving each differential coefficient. Letthese values ofthe differential coefficients ofybesubstituted inF=0;itthen becomes an equationwhich involves x,y,f,77and differential coefficients of i\ withrespecttof.Butwehave seenthat acandyarethesame as and77,since both setsarethecoordinates ofthesamepoint; therefore F= becomes adifferentialequationin17andonly. 80.Themost frequent exampleoftrajectoriesisthat in which asystemofcurves istobeobtainedcuttingagiven system ataconstantangle.Ifthisanglebearight angle,thetrajectory iscalledorthogonal;ifother than aright angle,thetrajectoryis calledoblique. Inthecaseoforthogonal trajectoriesthetangentsatacommon pointaretobeperpendicular,andtherefore which isforthis case theform ofF=0.Forthegiven system ofcurves wehave f(x, y,a)=0, dtsdydoc' fromwhichweeliminate aandobtain' a'relation betweencc,yand -f- ,which isreallythe differentialequationofthissystem of dec curves;letthisrelation be 80.] TRAJECTORIES. 121 Now forthetrajectory wehave j dyIand -*=-_ aceCLrj andtherefore thedifferentialequationofthetrajectoryis Theelimination oftheparameterisimmediate when theequa- tion ofthegiven familyofcurves occurs intheform <j>(OB,y)=a. Forwethenhave dscdydx' which atoncegives -^-independentofa,and istheform of^r= CwC forthis case. 81.When theequationofthecurve isgiveninpolarco- ordinates thesamemethod maybeapplied.Forwethenhave astheequationofthefamilyofcurves. If$hetheanglebetween theradius vector andthepartofthetangenttothecurve drawn from thepointback towards .the linefromwhich 6ismeasured, wehave while, if<E>bethesamequantityforthetrajectory,andRand bethepolarcoordinates ofapointonit, Since thetangentsareatright angles, 122 TRAJECTORIES.[81. andtherefore d6drfcRdR+l-Q' whereRand r,and6(butnot theirderivatives)arethe same. Now^+^^=0- dr36dr' eliminatingcbetween thisequation andtheequationofthe curve,wefindarelation oftheform Forthetrajectory T> r\ rt j&Q 1 Jt=r}=9,and--,-= J/tA= cZr-DidRdR thedifferentialequationofthetrajectoryistherefore This,whenintegrated, givestheequationofthesystemof curvespossessingtherequired property. Ex. 1.Find theorthogonal trajectoryoftheseries ofstraightlines Wehave ^=m> an(Jtherefore thedifferential equationofthese lines is dy*-* Hence, byourrule,thedifferential* equation ofthesystemoforthogonal trajectoriesis --'! which onintegration gives aseries ofconcentric circles havingforcommon centre thecommonpointof thelines. ' 81.]'TRAJECTORIES. 123 Ex. 2.Find theorthogonal trajectoryof Taking 1nga.rit.TrmH anddifferentiating, wehave n&r_cosnd rdo= sinnQ' which isthedifferential equationofthefamilyofcurvea Forthetrajectory wehave 1dr_..de rd6~~dR' andtherefore thedifferentialequationofthetrajectoryis ,, _ dR sinTie~ Thevariables maybeseparatedand dR sinTie,_n-5-=-n- -de,R cosnQ sothat .ft"=.4" cosw6, thefamily required. Ex. 3.Prove thatwhatever bethevalue ofntheorthogonal trajectory ofthecurves included in y=cxn isafamilyofconies. Ex. 4.Shew that theorthogonal trajectoryofasystemofconfooal ellipsesisasystemofhyperbolasconfocal withtheellipses. Ex. 5.Obtain theorthogonal trajectoryofthesystemofcurves (i)......rnsinn5=a"; (ii)......r3=a?log(ctan0),cbeing arbitrary. .Ex. 6.Shewthat,iff(x+iy)bedenoted byu+iv, where uand vare real, then thefamilies ofcurves u=const.,v=const., aretheorthogonal trajectoriesofeach other;andthefamilies ucosa+vsina=const.,fordifferent values ofa,areoblique trajectoriesofeach other. Inparticularshew that,ifv,soobtained, behomogeneousoforderTI,the thevalue ofuis "dv dvnu=x =y. ctyyox Howmaythevalue ofubefound whenniszero 1 Ex. 7.Findasystemofcurves cuttingataconstant angleother than right asystemoflfconcentric circles. 124 TRAJECTORIES.[82. 82. Ifoneofthevariables begivenasanexplicitfunction of theother andtheparameter,theequation-willbeoftheform y= tj>(x,a) ; instead ofeliminating awe<&ay proceedasfollows. Letthe equationoftheorthogonal trajectorybe 7=(f.)> where inthelastaistobeconsidered anunknown function of tobedetermined sothatthecurvemaybetheorthogonal trajec- tory.Wenowhave d#_3$ dxdx' d?)_d$ d4>da d%~d%+ fa,dj' andtherefore 30/369^ aA9f 9ad/ Now, asnofurther differentiations aretotakeplace,-wemay write~tmplaceof^-,sincexisequalto;hence wehave + ~Ul This isanequationbetween twovariables aand;when integrateditwilldetermine thevalue ofa,which, when' substi- tuted in 77=0(f, a), givestheorthogonal trajectory. Ex. Obtain theorthogonal trajectoryoftheellipaea represented by .y=a(!-*)*. Here --!--*, andtheequation determining ais 82.]MISCELLANEOUS EXAMPLES. 125 which gives Thisonintegrationleads totheequation a*(l-&=A-tf+ therefore theorthogonal trajectory requiredis MISCELLANEOUS EXAMPLES. Solve theequations: (ii)agj-a* <>^-'1 -*1-" , 1 (vii) y=!(3*1+a3 \/ /7*v2*? 2Assumingthattheprimitiveof isoftheformy=w+- ,provethat itisgiven by U=ABUI(.V+O) }v=Acos(a:+a) ~^~Obtain theprimitiveof 126 MISCELLANEOUS 3.Bythemethod ofvariation ofparameters deduce theprimitive of 4Prove thattheequation hasaparticular solution oftheform e**,provided (a&!-o^o)(aA-oA)=(a fl&a- andhence solve theequation, assumingthiscondition satisfied. (Schlomilch.) 6.Integrate .cPu . du sin2x -j-&+sinxcosx-j-=u. Ifu=Qwhenx=Qand?/=!when#=f,thenM=</2J-1when 5?=^.2 4 Also solve thedifferential equation determiningthearbitrary constants bytheconditions thaty=a and-/ when a;=0. 6.Theequations have asolution incommon; findtheprimitive ofeachandthenecessary relation between P,P1 ,Q,tysupposedtobefunctions ofx. 7.Prove thattheequation ;S-(**''i(a-l-fti(l-t-')'i(c-a- canbeintegrated bythemethod of68,provided therelation besatisfied forsome onesetofsigns given totheradicals. Find thesolution when thiscondition issatisfied. EXAMPLES. 127 8.Solve theequation wherea,b,kareconstants, byassuming y=(x+a)m(tc+b)n , andobtain thegeneral solution. Solvesimilarly theequation alsoEcs. 1in 68. 9.Provethat,if</>(#)beaparticular solution oftheequation cPx_ dx^=aX*' thena?0 (- )isaparticularsolution oftheequation Hence solve theequation d?z 10.Prove that ife= <f>(aa)beasolution of 7b^^^ ("^/i CL\Xt fax Ib\then=:(cx+d)<f>f^Jisasolution of theconstants ct,b,c,d being connected bytherelation adbo=l. Hence solve the firstequation inquestion8. 11.Shewhowtosolve theequation whereXisafunction ofxonlyand-4^ ...,Anareconstants. 12.Integrate theequation JTbeing anyfunction ofa;. -.. t*r''L .r'3'3 1 i^ ^t'^fT^' ^it 128'* MISCELLANEOUS L, f*^ 13.Shew that,ifaparticularsolution oftheequation beknown,^andJTabeingfunctions of#,theprimitivecanbeobtained. Hence solve theequation dy ,.asina; -^2 - dx 14.Theprimitiveof being y=A shew thatthedifferential equation which hasforitsprimitive -where (Hennite.) 15.Prove that,ifytandyabetwoparticular solutions oftheequation 3+*&*-* theroots of^=0and3/2=0separate each other solong asboth ofthese solutions remain continuous. (Sturm.) 16.Solve thedifferential equations: (i)sinB (Hi) EXAMPLES. 129 17.Solve theequation whereQandItsatisfytherelation When thisrelation isnotsatisfied, cantheequation hesolved bytheintro- duction ofafactorp.sochosen thatthenew coefficients satisfytherelation ? 18.Solve theequationi/r- **-- frCw'vdm*(Zcx-x*}*'L(Stokes.) 19.Findtheform of <such that, if#=$() besubstituted intheequation Z itwillbecome^ . andthence solve theformer equation. 20.Prove that'theequation -r^+-P-5^+<?#=0OBBbetransformed into when therelation between zandxisgivenby .and*(3)isgivenby Hence reduce theequation y^-- -jL=&U($-#*}totheform dPy 1dyf inF\_ 21.Solve theequation / .M/,,-if whereAand5areconstants. Verifythattheequation istransformable intotheforegoing equation bythesubstitution provided #2=45i! andfindtherelation between yand v.Hence solve thesecond equation. F. 9 ' fi "J_i. ft-A' 13Q-U- MISCELLANEOUS ,.,,,^ 22.Bytransforming thedependent variable fromyto&,solve t&e equation Hence solve theequation S^M^S'-i&fH^-^-'S}- 23.Prove thattheprimitive oftheequation fflo- 5da\* where a-istheSchwarzian derivative ofywithregard tox,is y(A1+E'ss+C'x*)=A+Bx+Cx*', andshew that this isalsotheprimitive of B>3y2,3ft=0, whereyl}yz,...arethefirst,second differential coefficients ofy. 24.Prove thattheprimitive of is wheren=l-2A Discuss thecase inwhichd,supposed constant,isequal 2* 25.Thearcofaplane curve measured fromafixed pointAuptoapointPwhoserectangular co-ordinates arexandyisdenotedbys-obtain the general Cartesianequations ofthecurves forwhich thefollowing equations respectively hold : (i)=(^+y2)*;(ii) a= (V)^+^^=0; ^)^C^+affiB)*; (vii),= 26.Find thegeneral differentialequation ofallparabolas touching theaxesandhaving their chord ofcontact ofconstantlength. Solve theequationobtained. Obtain alsothedifferentialequation ofallparabolas touching theaxes. 27.Shew thatthedifferentialequation ofageneral conic is andofageneral parabola is (Monge andHalpheiL) EXAMPLES. 131 28.Find(i)thecurve inwhich theradius ofcurvature isproportional to thearcmeasured from afixed point ;(ii)thecurve inwhich theproduct of theperpendiculars fromtwofixed points onthetangentisconstant;(iii)the curve which hasanevolute similar toitself. 29.Find adifferentialequationofthe first order ofthecurve, whose radius ofcurvature isequaltontimes thenormal;andshewthat itisalways integrable wheiinisaninteger. Inparticular shew thatwhen TI=2thecurve isacycloid, whenn=lacircle, whenn=-1acatenary. 30.Shew thatthesystemofcurvescuttingataconstant angle aother thanright asystemofconfooalellipsesisgivenby where 2oisthedistance between thefociandnistan a. (Mainardi andMukhopadhyay.) 31.Obtain theorthogonal trajectoriesofthecurves (i)aP+y*=ca!- t (ii)a?+y*+<?=l+2cxi/ ; (iii)aP+ya=3aay; (iv) rr'=c?; inthelastrand r'arethedistances fromtwofixedpoints. 32.Thecurve forwhich theordinate andtheabscissa ofthecentre of gravityoftheareaincluded between theordinates x=aandx=x areinthe same ratio astheboundingordinate yandtheabscissa xisgiven bythe equation 33.Thecurve whose polar equationisrncoamd=tfnrollsonafixed straightline. Assumingthatstraightlinetobetheaxisofor,shew thatthe locus ofthecurve described bythepoleintherolling curve willhave for itsequation2m Inparticular shew that,when2m=1,thedescribed curve isacatenary; whenm=2 thedescribed curve isanelastica. (Frenet.) oPy 34.Shew that,when afirstintegraloftheequation jj=/(#y)isgiven intheform-5?=ty (so,y,c),thentheprimitiveis \\ (Jacobi.) AfirstintegralofJ|=y(I+2tan2x]isoftheformir=y$ (#) determine theprimitive.92 CHAPTER V. INTEGRATION INSERIES. 83.ITmayhappenthatadifferentialequation,thesolution of which isrequired,comes under none oftheprecedingclasses which areallofsomeparticular form, andtherefore that themethods applicabletothese fail;recourse istheuhadtoapproximationto obtain thevalue ofthedependentvariable. Theform ofapproxi- mation which ismostfrequently adoptedisthat derived from convergingseries;byretainingalargenumber ofterms theerror canbemade small, andtheseriesmaybeconsidered tobethe value ofthevariable. That thismethod is&priori justifiable maybeseen asfollows. Thegiven equationisarelation between thesuccessive diffe- rential coefficients ofyandmaybeconsidered asgivingtheone ofhighestorder interms ofthose oflower orders;thus ifitwere ofthesecond order itwould give-^interms of-r-and y.When dor dxy differentiated once itwould give~.interms of-T^S ,-rand v,dxador dooff> thatia,interms of -j-andy,since^isexpressibleinterms ofthese two,Hand soforeach ofthe differential coefficients of higher order, which canthusbeexpressedinterms of-^-andydxy' butthe differentialequationwillnotgiveanyrelation between 83.] INTEGRATION INSERIES. 133 dii -pandy,which arethusindependentofoneanother.Suppose nowthatavalue abeassignedtoasandthat forthisvalue ofsowe diimakey=Aand^=-#>which constants are,ingeneral, arbitrary; thentheequationsderived bysuccessive differentiation furnish the values for as=aofthedifferential coefficients ofyofsuccessive orders. Letthese bedenoted by0,D,JS,....Now ifthevalue ofybe(as),whichweassume isafunctionexpansible byTaylor's theorem inaconvergingseries ofascending powersof as a,we have i(g-a)'d'^(a) ,(*-offfj>(a).+-TT--35r+___ +..., wherej;'stands forthevalue of^-^when aiswrittenda CM; for CDafter differentiation.Inserting now forthevarious coefficients their values, weobtain andthis, ifaconverging series, isasolution ofthegiven equation. Itshould beremarked that forsomeparticularvalue ofocthe differentialequation maydetermine notthecoefficient ofhighest order butoneoflower order;thustheequation d?y 2ndy_ da? xdxy~ d*vwould forvalues ofa;other than zerodetermine -r4 ,butfor as= dor wouldgive-~-=0,ifweconsider infinite values ofanycoefficient excluded. Theforegoing method andanother, which isin*practicesubsti- tuted foritandwhich willbeexplainedinthenext article, is almostimpracticableinthecase ofequations whicn neither are linear norcanbetransformed soastobecome linear; forsuch equationsthedetermination ofmore than the firstfewterms of theexpansionentailsgreatlabour. 134 INTEGRATION [83. Ex. 1.Letusapplytheforegoing method totheequation When differentiated ntimes theequation gives andtherefore when#=0 x Nowthegiven equationleaves yarbitrary, say=A,and^arbitrary, say=5, when#=0;but-^L=0. "?o Hence wehave =(-!} =0; similarly and Theexpansionofyis,byMaolaurin's theorem, This isthesumoftwoconvergingseries andcontains twoarbitraryconstants; itisthustheprimitiveoftheequation. Ex. 2.Solve (ii) 83.]INSERIES. 135 Ex. 3.Obtain anintegraloftheequation s+ intheform _..wu;mzaP~" I5PT2ala .2a .3aI2 .22 .32 .4s~ 84.Theprecediag investigationshews that,bymeans ofthe differentialequation andtheexpansionofafunction interms of theindependentvariable asgiveninTaylor'sorMaclaurin's theorem, anexpressionintheform ofaseries canbeobtained for thedependentvariable;but, instead ofworking throughwhat is sometimes atroublesomeprocess,itisconvenient toacceptthe principlethataseries canbeobtained andsotoassume fory some seriesarranged accordingtopowersof<Kwith indeterminate coefficients and indices. This series isthen tobesubstituted for thedependentvariable inthedifferentialequation, andasitisa solution ofthatequationitmustmate theequation anidentity; acomparisonoftheindices oftheindependentvariable will shew thelaw oftheirprogression,andacomparisonofthe coefficients ofthedifferent termsinvolvingthesamepowersof thevariable willgivetherequiredrelations between thecoefficients intheexpressionassumed. The latter willthen forsuch values of theindependentvariable asleave the seriesconvergingbea solution. 85.Asthemethodjustindicated isreally equivalenttothe earlier one,itisnotbetter suited tothesolution ofnon-linear equations ;butmuch labour issavedbyitwhen the differential equationtobesolved islinear. One ofthemostimportantforms towhich itisspecially applicableisthatwhichmaybewritten where andtyarerationalintegral algebraicalfunctions. To solve this,assume y=A^+AjF*+AtaT+..., wherem1}raa,ms,...areexponentsinascendingorder ofmagni- tude; since 136 INTEGRATION[85. theequation,-withthevalue ofysubstituted init,gives (mja5>+Az<$>(mja;"1'+ ... +...=0. Inthisequation m^1isthelowest exponent and itoccurs in onlyasingleterm;astheleft-hand side istovanishidentically, thistermmustdisappear,andtherefore or,sin#eA^isacoefficient ofatermactually occurringandsois notzero,wemusthave Acomparisonoftheindices oftheremainingterms shews that ml=ma1andtherefore m2=mt+1, andsoon;while acomparisonofthecoefficients ofterms involv- ingthesame indicesgives and soon.Takenowanyvalue ofm1asgiven bytheequation i/r(mj)=0,saym^=a,;then asA^isquite arbitrary denote itby A.Theremainingcoefficients aregivenby ._ 8 8"1" ' andsoforthehighercoefficients;thecorrespondingvalue ofyis thus 0+,/ y-(a+i;Y^a-i-z; rk(n\Afn j-~\\Afn -L.V\ a"+.... 85.]' INSERIES. 137 Theexpressionsconnected with theother rootsmaybesimi- larlyobtained;andastheequationislinear thesum ofallthese values ofyisasolution, Ofthisgeneralformthemostimportant exampleisthatequa- tionwhich hasforasolution theseriesknown asthehypergeome- tricseries;itisdiscussed infulldetail inthenextchapter. Ex. 1.Prove thattheprimitiveoftheequation cPy ,2?z^+7 isgivenby 1"' \ Ex. 2.Inthecasewhen2=1,theseparate parts involving thearbitrary constants inEx.1become thesame, eachbeing Ifthisbedenoted byv,andyuv=w twhere uandwaretobedetermined, wehaveonsubstituting,since visasolution oftheoriginal equation, cPw 1dw Aswehavetwoarbitrary quantities uandw,wemayassign anyonecondition weplease ;letthisbe cflu 1du^ft ax*xdx Thevalue ofuhence derived isA+Blogx,andthus dho ,1dw , ,ZBdv _ T-U+--j-+mw-\--j-=0,dx*xdx xdx' or d*wldw Thevalue ofyisnow andtherefore contains twoarbitrary constants, thetotalnumber necessaryfor theprimitive ;hencewerequire onlyaparticular integraloftheequation inw.Toobtain thiswrite -INTEGRATION[85. then Substituting andequating coefficients ofdifferent powersofx,wehave from the coefficient ofa?-1.........Bl=Q; ............... afl .........m These equations give ^=0=53=... --3-ir-. sothatnotermsinvolving oddpowers ofxoccur inw.For. -the coefficients ofevenpowers wehave m- 2a.4a . andgenerally Hence thevalue ofyis fnmtf2-_ i, ,p,,2 ,m2^mW\ +2+2:'.4'1~21!.4a.62+ '"J As5*isundetermined, there areapparently threearbitraryconstants; but itwillbeseenthattheexpression multiplied by'isthesame asthat multiplied byAandtherefore these twoconstants coalesce intoonenew arbitrary constant A'whichmayreplace A+'. 85.] INSERIES. 139 Ex. 3.Obtain theprimitiveoftheequation (Fourier.) Ex. 4.Integrateinseries, andexpressinafinite form, theprimitivesof thefollowing equations: 86.There aretwospecial pointswhich arise intheintegration ofsome differentialequations ;theyowe theirorigintothesame cause, butthey requiretobedealt withseparately. Asanexampleoftheone, letusrecur totheseries obtained as asolution oftheequation which was theconstant abeingsome root oftheequation i/r(m)=0. Thisequationwillusuallyhavemore than oneroot;letsome other rootbedenoted by6.Then, inthecasewhen 6isgreater thanabysomeinteger k,thesolution intheformabove adopted ceases tobeavailable;forinthedenominator ofthecoefficient of 0*within thebracket there occurs thefactor-\Jr(a+k}or-^(6) which iszero, sothat, unless there beazero factor inthenumera- tor,thecoefficientapparentlybecomes infinite. Inthecasewhen such azero factor does notoccur inthe numerator wemust have recourse tothefundamental equations fromwhich theseries wasderived, which are 140 INTEGRATION[86. =0. Now since^(a+ft)vanishes andAMisnot infinite, beinga coefficient inaseriessupposed converging,itfollows that either Akor <J3(a+k1)iszero.Rejectingthe latter onaccount ofthehypothesisthatnozero factor occurs inthenumerator wehaveAt=0,andthence from thepreceding equations we findthatthecoefficients -4];Aa,...,A^are allzero. Hence the partoftheseries whichprecedestheterm #*inside thebracketis, onaccount ofitscoefficients, evanescent, andtheseriesactually mustbeginwiththeterm Coca+k ,thatis,withCxb ;and thiswillbe theseries derived from theroot boftheequation ^(m)=0.One oftheparticularsolutions hasthusdisappeared,buttoobtain one initsplacewemayproceedasinEx.2in86.Denoting byv theonewhich remains andhasabsorbed theother,wemaywrite and, after substitution, assignsome onerelation which shall serve todetermine uandwandrender thedifferentialequationeasier tosolve;this relation willusually bedeterminedbythespecial form oftheequation. Ex. 1.Consider thedifferentialequation Substituting y (thisiseasilyseen tobethenecessary form), wefindastheequationdeter- miningm TO(MI l)-4m+4=0, ie.(m-l)(m-4)=0. Hencea=land6=4, sothattheroots differbyaninteger.Itwillbefound that,ontakingtherootm=l, theequationisoftheform discussed andthat theterms upto,butexclusiveof,a?4disappear; while theseries derived from' therootwi=4 isAx*(P. Completethesolution. Ex. 2.Solve 87.]INSERIES. 141 87.Wenow proceedtoconsider theotherspecial point. Hitherto ithas"beenassumed thatnovanishingfactor occurred in thenumerator;andtheresult ofthenecessaryalternative was indicated. Butavanishingfactormayoccur inthenumerator of some ofthecoefficients oftheterms within thebracket, either in that terra inwhich there isavanishingfactor inthedenomina- tororinanearlier term. Inthelatter case alltheterms which donothaveavanishingfactor inthedenominators oftherespective coefficients disappear ;and ifsuch afactor never occurs inalater term theseries willendattheterm next before the firstwhich contains thatvanishingfactor inthenumerator, andthesolution willthusbeexpressedinafinite form. Butsomevanishing factormayappearinthedenominator ofalatertermandtheco- efficient ofthisterm willthen taketheindeterminate form0/0, while theinterveningterms willdisappear; and alltheterms after this willcontain thisindeterminate coefficient. The series will thenbeoftheform where k1isnotlessthan/.Thismaybewritten whereAisarbitrary andB/A, ...,F/Aaredeterminate;M,being equaltoKx0/0,isarbitrary (onaccount oftheindeterminateness of0/0)andLjK,...aredeterminate. This series isasolution ofthecorrespondingdifferentialequationandtherefore willbea solution when aparticularvalue issubstituted forthearbitrary constant;hence obtained bywritingM=0,isasolution. Insuch acasethere is therefore asolutionoftheequation expressibleinafinite form. A Ex. 1.Consider asanexample 142 INTEGRATION INSERIES.[87. When wewrite y=Atfm+Bafm+1+..., theequationtodetermine mis m2-9=0, andtherefore m=-3or+3. Fortheroot 3itisnotdifficult toobtain theseries r22i ~i Aa;~s \I--x+=-L-sJ<P+terms inof,at,x6which vanish |_5 5.a J -2.-1.0. 1.2. 3 . -2.-1.0. 1.2. 3.4 . WriteMinstead of -2.-1.0. 1.2.3 -6.-8.-9.-8.-5.' andthentheseries is 4. 4.5 4.5.6 ^+7(4*-3a )(5s1-3") (4s-3s )(52-32 )(6*-3s ) ...], thusverifyingthetheorem thatonesolution oftheequationisexpressiblein afinite form. Ex. 2.Verifythegeneraltheorem inthecaseoftheequation Ess. 3.Solve theequation 88.Further illustrations ofthesespecial pointswilloccur later andtheyneed nottherefore nowbeconsidered ingreaterdetail; various otherpointsarisewhich willhediscussed inconnexion with special equations.Thus ithasnotheen stated thataseries must always proceedinascendingorindescending powersoftheinde- pendent variable, butthecomparisonoftheterms inthedifferential equationafter theexpressionforthedependentvariable hasbeen therein substituted willindicate thenature ofthe series. Inthe casewhen oneofthesolutions becomes evanescent onemethod has beenpointed out,which willbeuseful forsupplyingthedeficiency thus caused; another willbeindicated below. Infactthedifficul- tiesthat arise areusuallyconnected withspecial equations andnot 88.] LEQENDRE'S EQUATION. 143 with thegeneral equation; andtherefore somespecial equations willbeconsidered. Ofequationsofaparticular form there are fourwhich aremoreimportant thantheothers included intheclass soluble inseries;theyare First, the differentialequationofthehypergeometric series which willbediscussedseparatelyinthenextchapter ; Second, Legendre's equation ; Third, "Bessel'sequation; Fourth, Kiccati'sequation. The lastthree ofthese willnowbediscussed inorder. Itmust ofcourse beunderstood thatwhat iscarried outhere ismerelythe completesolution ofthedifferentialequations andthat there isno attemptatanexhaustiveinvestigationofthepropertiesofthe respectivefunctions determined bythedependentvariables. LEGENDRE'SEquation. 89.This differentialequationis or,what isthesameequation, inwhich thequantitynisaconstant. Theequationisonewhich frequentlyoccurs ininvestigationsconnected withquestionsin most ofthebranches ofapplied mathematics;inthese cases nis usually,butnotalways,apositive integer. Theequationisone ofthesecond order andhastherefore twoindependent particular solutions, andeveryotherparticularsolution canbeexpressedin termia ofthese two;but itwillbefound thattheform ofthese fundamentalparticularsolutions isdifferent inthetwocaseswhen nis,andwhennisnot,apositive integer. Weproceedtoobtain these solutions. Inaccordance withthe generalmethod ofintegration byserieswewrite y=A 144 LEGEN'DRE'S[89. andsubstitute;thenwehave n(n+1)(Ajtc^+A^+AjF*+ ...) ={(a?-1)(XAX^+m^X1'-1+m^/V1+...)} =ml a(ms+1)AjF*-m3(ma- and thismust "beanidentity. Aninspectionoftheequation shews that, sofaraspowersof#areconcerned, wehave mi=m l-2, w8=m9-2, ortheseries must beoneindescending powersofx;wethere- forenowassume thatm t,m4,mg)...arearrangedindescending order ofmagnitude,theircommon differencebeing2.Acom- parisonofcoefficients ofthesamepowersofasgives,forthose ofa?1"1 , [vo^(m,+1)-n(n+1)}A^=0, or NowA1isnotzero,beingthecoefficient ofthehighest term in y;hence either 77i,=n, orTTij=(n+1). The relation between the coefficients ofconsecutive terms arises fromequatingthecoefficients ofa5mi~lr+aonthetwosides;itis,for values ofrgreaterthanunity, 71(Ti+1)Ar=(m t-2r+2)(TTI,-2r+8)Ar -(mt-2r+4)(m,-2r+3)Ar_lt .and thisgives* 90.Consider firstthesolutioncorrespondingto m,=n. 90.] EQUATION. 145 Thehighest term isthenAjp ;andtherelation between thesuc- cessive A's is sothat A '2(r-l)(2n-2r + 2^.1. 2.3...(r andtherefore theseries becomes 2(2n-l) 2.4(27i Lettheseries within thebracket bedenotedbyyi}which is therefore aparticularsolution. When nisapositive integer, the series isfinite;thelasttermis,when niseven, 2.1 or,what isthesamething, nlnlnl while, whennisuneven, thelastterm is /_l\*(-D^ n(n-T)(n- 2) 3.2 > ' n A. t o\f~, or,what isthesamething, thenumbers ofterjns inthetwocases arerespectively %n+1and When n,isaninteger,2?iisaneveninteger,aftd*therefore a zerofactor cannever enter intothedenominator inthiscase;thus theseries considered willnever come under theclass considered in 87whichyieldstwosolutions. -F. 10 146 . LEGENDRE'S [90. The series yl}multiplied by 2" .n In ! nbeingapositive integer,isusuallydenoted byPn;thisfunction isanextremely importantoneinphysical applications. Ex. 1.Verifythat andthatPwisthecoefficient ofenintheexpansioninascending powersofz of(1-asa+s8)-*.})w %g^f-dO-LJLj6""2-P Hence shew thatv=(1-Ste+a2)"*isasolution oftheequation 3a* 9 .&. 2.Prove thattheroots oftheequation 7^=0areallrealandnumeri- callylesshanunity. Ex. 3.Prove thatthesum ofthecoefficients inPnwith theirprqper signsisunity. ,***&. c^r.j.,Jtwt^tC. ,=,')_ -*)'ll^-*f"' nt (i),***&. c^r j., &i.Vjud<U ^^ r.4Obtain theequationsjV"iv^ Inthecasewhennisnotapositive integertheseriesy^pro- ceeds toinfinity;and forconvergenceitisnecessarythatarshould begreater thanunity. Butinparticular when 2wisequaltosome positive oddinteger, say2r-1,then thecoefficient ofxn~*hasa zero factor inthedenominator, andnozero factor occurs inthe numerator either ofthatterm orofanysubsequent term;hence (by.86)theterms whose indices arehigher thann2rdonot exist inthissolution ofthedifferentialequation, which willthere- forebeginwithaT*multiplied bysomenewarbitraryconstant. Butsince 2?^=2r1,therefore n2r= (71+1),orthesolution degeneratesintoaninfinite series ofdescending powersofoc beginningwith af(n+1) .Totheconsideration ofthis solution we shallnowproceed 148 LEOENDBE'S[92. 92.Wethushave thefollowingresults. I."When nisapositive integer, there aretwoindependent solutions ofthedifferentialequation;(1)yltafinite series, (2)ys, aninfinite aeries;andtheprimitiveis II.When nisanegative integer,there aretwoindependent solutions; (1)yi}aninfinite series, (2)yvafiniteseries; andthe primitiveis IllWhen nisnotintegral andZnisnotequaltosomeodd positiveornegative integer,there aretwoindependentsolutions; (1) 2/1}aninfinite series, (2)y3,aninfinite series;andtheprimi- tive is IV.When 2wisequaltoanoddpositive integer,there has been obtainedonlyonesolution ofthedifferentialequation,fory degeneratesintoya,this solutionbeinganinfiniteseries; the primitiveisthusnotexpressibleinterms ofytandy^alone. V.When 2nisequaltoanoddnegative integer other than I, there hasbeen obtainedonlyonesolution ofthe differential equation,foryadegeneratesintoy1,thissolutionbeinganinfinite series;theprimitive againisnotexpressibleinterms ofy^andya alone. YI.When 2nisequalto 1,there hasbeen obtainedonly onesolution ofthedifferentialequation,for7/aandyaarethesame infinite seriesbeginningwith aj~*;theprimitive againisnot expressibleinterms ofy1andytalone. Ittherefore remains toobtain theprimitive inthelastthree cases. 93(i).Consider firstthecaseof%nequal toanoddpositive integer ;then 2(2a+3) isadefinitesolution, andwehave tofindasecond and differentparticular solution. Inthefirstinstance, assume 93.] EQUATION. 149 where 6isaninfinitesimalquantity which willultimately bemade zero. Then, solongas6isnotzero,thequantity isalsoadefinite solutionjand itceases tobesobythevanishingof6,since 9enters asafactor intothedenominator ofthecoefficient ofaP~*&~* and all lower powers. Nowwehave ^ 2.4...2p(S-l)(2-8)...(2*-2p +l) -2p-l)__ ^.2 I J +(1 ^2.4...2j3(2?t-l)(2n-3)...(2w-2p +l)a?n j ta"^-a f(n-2p-2)(n-2p-3J^, | r 2tt-2jD-l 1 (2jo+4)(2i-2^-3)Tl "J' where andsoisdeterminate and finite. But n-2p-2- andtherefore Alsothecoefficient ofa>~arwithin thesecond bracket is (tt-2j3-2)(m-2p-3)...(ft-2/?-2r-l) ^^(2p+4)(2p +6)...(2j>+2r+2)(27i-2ju-3)(27i-2jo-6)...(2?i-2p-2r-l)' i.e.is (-1)' i.e.,is 91.] EQUATION. 147 91.Wetakenowthesecond solution oftheequationdeter- miningthevalue ofm,^;this is (n+1),sothat theterm with highestindexmaybetaken tobeAjc~(nH] .Therelation between thesuccessive coefficients is (271+2r- 1)(2r-2)Ar=(n+2r-3)(n+2r-2)A^ forvalues ofrgreaterthanunity, andtherefore A_ |-~- 1 .(2+2r-l)*' sothattheseries is 1-1*11 |( *\a+2(2+8)AA Lettheseries within thebracket bedenoted byya,which isapar- ticular solution; theseriesyamultiplied by W.nlnl (271+1)1 (nbeingapositive integer),isusuallydenoted byQn;forcon- vergenceitisnecessarythat a;should begreaterthanunity.This seriesya,ortheequivalentfunction Qn,isalsoofgreat importance inphysical investigations. When nisapositive integer,theseriesproceedstoinfinity.' WHen nisanegative integer, yaisafinite series;ifn=2p, theseriesbeginswith tf8^1andproceeds for^jterms;ifn=(2p+1), theseriesbeginswith#*andproceedsforp+1terms. When 2nisequaltoanoddnegative integerother than1, say (2r+1),then thecoefficient of#-fn+Br+1>hasazero factor in thedenominator, andnozero factor occurs inthenumerator ofany term intheseries; hence asbefore theprecedingterms donot exist andthe seriesbeginswith af(n+Sr+' 1)multiplied bysome newarbitraryconstant. But since 2n=(2r+1),therefore (n+2r+1)=n,orthesolutiony^becomes aninfinite series of descending powersofxbeginningwith #",i.e.ysdegeneratesintoy1. 102 150 SPECIAL OASES OF[93. where s=r 1 t=>r 1 8=2i- I_2_i2 2r Hence n(n-l)...(-% +l)I '2.4...2n(2-l)(2n-3)...(2-2p +l) J fi+T-f Lr=il2.4... Wlien thesecond partoftheright-handaide isexpandedtheaggregateof terms which involve -is^ya;theaggregateofterms which involveloga?is and there remains theaggregateofterms independentof6(and also asit appearsoflogx\aswellasafurther aggregateoftermsmultiplied bypositive powersof6,most ofwhich havebeenomitted and allofwhichdisappear when 6ismade tovanish. From thefirstpartoftheright-hand sidethere isan aggregateofterms independentof6,aswell asanaggregateofterms which disappear when 6ismade zero. Hence theprimitiveoftheequationis onchangingthearbitraryconstants. HereTnstands for '^ f (-!) a fl\2(2-ir +f_iyn(-l)...(n-2p +l) J ^ ;2.4...2p(2w-l)(2?i-3)...(27i-2p +l) J' andEnstands for ---f(n+l)(n +8)...(n+Br)1 r=i\2.4...2r(2?H-3)(27i+5)...(27i +2r+l)^ J' thevalue ofCvbeing =r Thevalue ofthecoefficient A/0which occurs inTnisu 93.]SOLUTION OFLEGENDHE'S EQUATION. 161 sothatwemaywriteTnintheform Thesecond particularsolution oftheequation isthus and itwillbenoticed thatthatpartofitwhich isexpansibleindescending powersofxbegins withaterminvolving x+nandcontains noterminvolving Butinthespecialcasewhen2%isequaltounity,sothatpiszero inthe preceding investigation, then theform ofTn,nowT\say,islimited tothe firstterm; andwehave 0A*%3-VL. BOthat Theremaining partsareunchangedinform. 93(ii). Consider nowthecaae of2raequaltoanoddnegative integer other than-1;theintegral yisdefinite, but willthennotbeadefinite solution. Before assuming ntobehalfanoddinteger, write (sothat2misapositiveoddinteger when theassumption astothespecial value ofnismade). Then 2(2m+3) whereFtandF2arethespecialsolutions of ^The solution thus given correspondstothat forBessel's equation,Ex. 1, p.167,duetoHankel. 152 SPECIAL OASES OF[93. mbeing positive. When2misanoddpositive integer weknow from the preceding investigation thattheprimitiveofthis is where and A.-^riTl(+l)(+2)...(+ar) ,J P=il2.4...2r(2f+3)(2i +S)...(2nH-2r+l)*^ J* thevalue ofArbeing Hence theprimitiveof inthecasewhen 2%isanoddnegative integer other than-1,is y=B^+A (Vllogx+Vn+Un\ where 2(271-1) 2.4(2-l)(2n-3) and where inUnthevalue of.EJ.is . Thesecondparticular solution oftheequation inthiscase isthus and itwillbenoticed thatthatpart ofitwhich isexpansible indescending powersofxbegins with aterminvolving ar""1andcontains noterm in- volving#". 93.]SOLUTION OFLEGENDRE'S EQUATION. 153 93(iii). Lastly,forthespecial caseinwhich Znisequalto- 1,weproceed inamanner similar tothatadopted in93(i);andwefindthattheprimitive oftheequationis whereytistheseries and i,4r-l and 94.Since inallthese cases 2wisanoddinteger, theequation canbe writtenA-2TY_ I_H-J--!^ r ,=iV2>'+2-1^2a-l 2s,T wherepisaninteger. Thecaseofppositiveisthatconsidered in93(i);thecaseofpnegative isthatconsidered in93(ii) ;andthecase ofpzero isthatconsidered in 93(iii). Properties ofthefunctions defined bythedifferentialequationin thepresent form have been discussed byMrW.M.Hicks inhismemoir on "Toroidal Functions," Phil. Trans.Roy.Soo.(1881), pp.609652. Ex. 1.Assuming theresult ofEx. 1in64,shewhowthesolution of canbederived from that of which isthedifferentialequationforthequarter-period inelliptic functions. Ex. 2.Prove thattheParticularIntegraloftheequation isX-P,^, where Xisaconstant; andthat theParticular Integral ofthe equation" where X'isaconstant. 154 SPECIAL CASES OF[95. 95.Inthegeneralcase ofthedifferential equation, asrepresented byI., II.,III.of92,itispossible toexpressthesecond particular solution interms ofthatalreadyobtained andofsimilar functions. Letvdenote theparticular solutionalready obtained, sothat forinstance vwould bePninI.;and let y=uvw} where uandwareasyetindeterminate. When this issubstituted inthe differentialequation, wehave Since visasolution, thelastterm disappears ;and,astheonly condition imposedonuandwisthatymustsatisfytheequation, wemay arbitrarily assignanother. Choosingthissothat thecoefficient ofvmayvanish, wehave andtherefore (sc21)-^-=constant.x 'dx Asweareseeking aparticular solution,itisconvenient tohave itas simpleaspossible ;andtherefore, giving aspecial value totheconstant, we maywrite sothatavalue ofuisgiven by Theequationtodetermine wnowbecomes d( ,.dw] , , ,. ndv When theParticularIntegral, saywl}ofthis isobtained, thesecond solution oftheoriginal equationis Thevalue ofw^asaseries ofdescending powersof%iseasily obtained. Thus inthecasewhennisapositive integer wetake a.4(a-l)(3n-3) andatoncehavetheequation, which determines w1}intheform 95.]SOLUTION OFLEGENDRE'S EQUATION. 155 Let MJ=Cja^1+C'j-r"-9+C^-6+... then, substituting andequatingthecoefficients ofthehighest term,wehave Cj,{n(n+1)-n(n~1)}=2n, orCi-lj andequatingthecoefficients ofthetermsinvolving #n~2r+1 ,wehave 2.4...(2r-2)(2n-l)...(2n-2r+3)' Thegeneral value ofCr}deducible fromthia,iscomplicated ;thevalues of theearlier coefficients are 33(2n-l)(2n-2)' c=(n-l)(-2)(tt-8)(n-4)(SOw2-50n+12) 33.4.6(2n-l)(2n-2)(2-3)(2-4)' andsoon;butthere isnoadvantageinwriting downmore ofthecoefficients, astheexpressionforw^willsoonbeputintoadifferent form. Relation between theparticularsolutions. 96.Wehavenowobtained theprimitive ofLegendre's equation inall caseswhennisarealconstant, bydeducing twosolutions which arelinearly independent (72)ofoneanother. Butweknow(65)thatwhen onesolution ofadifferential equationofthesecond order hasbeen found, theprimitive canbeexpressedinterms ofitand,ifnecessary,ofotherfunctions, and therefore anyother solution issoexpressible; weproceedtoobtain this relation forthecases viz.I.,II., III.above inwhich ithasnotbeen obtained. The firstform inwhich itmaybegivenisderived bymeans of 65.WemaydefinePnandQnbythegeneralised equations a*n()n(n)fand y"~n(2n+l) r whether nbeintegralornot;n(n)isGauss'snfunction and isr(ra+ 1),and inthecase ofnintegralisn 1(seenextchapter, 126) ;andPnandQnare stillsolutions oftheLegendre's equation,since theyarerespectivelyconstant multiplesofyxandyz.Wetherefore have 156 RELATION BETWEEN PARTICULAR[96. multiplying theformer byQnandsubtracting thelattermultiplied byPn, wehave whereAisaconstant, which isdefinite andnotarbitrarysinceQnandPn aredefinite functions. TofindAweconsider thetermscontainingthe highest powersofx;these are 2n(n)n(n) ( andinPn hence A sincen(2a+ 1)=(2^-+ 1)n(2?i) ;andtherefore *Vndx Thisgives or,itsequivalent andtherefore noconstant being needed, asmaybeseenbycomparing thecoefficients of thehighest powers ofxintheexpansion ofthetwosides indescending powers ofic. 97.This resultmaybewritten inadifferent form; but itisfirst1neces- sarytoprovetworelations between thefunctions given byLegendre's equation fordifferent values ofn. From theexpressions given intheprecedingarticle* wefind thatthe coefficient ofa"+1-*1inPnt-P1l_iB f(2n { thelastfactor iseasily simplifiedinto 97.]SOLUTIONS OFLEGENDRIi'S EQUATION. 157 andtherefore thecoefficient is Hence thecoefficient ofa;"~2rin dx dx n()2.4...2r(2n-l)(2n-3)...(2n-2r+l)' thatis,isthecoefficient ofthesame powerin(2n+l)P n.These two ex- pressionsarethusequal termbyterm;andtherefore or Inthecasewhennisapositive integerthisleads toafinite series for dfviz~3s' thelastterm oftheseries 3FiorPO(i-e- 1)>accordingasniseven orodd. If%benotapositive integer, theseries willproceedtoinfinity and will *fJP stillbethevalue of-,- ,provided#begreater thanunity.dx 98.Nowby95weseethat isasolution ofthedifferential equation,ifwbedetermined astheParticular Integralof bytheformula justobtained. Toobtain thisParticular Integral wewrite w=a1Pn andsubstitute;since ' theleft-hand sidehas,asthecoefficient ofa2r-i andtherefore oSr'_1(ar-l)(n-r+l)=2- 158 LEGENDRE'S EQUATION. [98. Thevalue ofIDistherefore now definite; andthecorrespondingsolution ofLegendre's equationis thelasttermbeing 3p(-l)(KH)" whenniseven,and '1 "n, i-e., when 7iisodd. 99.Wehavenow tocomparethissolution withQn.Let ithesupposed expandedinaseries ofdescending powersofx;itmust thenbeoftheform whereAandBareconstants. Now intheseries theterminvolvingnondoes notoccur,since _1 _1_ _1_ andtherefore Amust bezero;hence thecoefficients ofthepowersbetween a?1and~*n+1)exclusive ofthelatterdisappear;this iaeasilyverified for thefirst few.Theabove solution istherefore aconstantmultipleofQn,and thus whereZnstands fortheseries which, whennisintegral,isafunction of degree n 1.Hence andtherefore where Z7isanintegral function ofxofdegree nothigher than 2ra- 2.When wesubstitute ontheleft-hand sidefrom 96,itbecomes B IU ^~nT^T-nnJ or = where theright-handside isafiniteintegralfunction, ofx.This istrue for allvalues ofx;writing x=lwehave5=value ofPnzwhenxisunity. Now inEx.1of90,Pnwasindicated asthecoefficient of^intheexpansion of 99.]BESSEL'S EQUATION. 159 (1-tocz+z*)'* inascending powersofzandtherefore thevalue ofPnwhen #=1 isthecoefficient ofznintheexpansion of(12z+)~b, i.e.,of(1-s)-1 . This coefficient isunity,sothatPnwhenx=\ isunity ;thusj?=landthe equation becomes 17.I.Thefollowing properties, analogous tothose ofPn,hold forQn: (i) .fik. 2.Obtain thepropertiesoftheintegrals Qcorrespondingtothose of theintegralsPgiveninEx.4,90. Ex. 3.Prove that, ifasbelessthany, Thefurther developmentofthepropertiesofthefunctions which arethe particularsolutions ofLegendre's equations doesnotdepend merely upon the differential equation ;thestudent willfindmostample investigation oftheir analytical properties andtheirapplications tomathematicalphysicsinthe excellent treatise byHeine Handbuck d&rKugelfunctionen. The treatises byTodhunter, TheFunations ofZaplace, Lam4andJBessel, andbyFerrers, Spherical Harmonics, willproveuseful.' BESSEL'SEquation. 100. This differential equationis or,what isthesamething, inwhich nisaconstant;itwillbeassumed that nisreal. Theequation,likeLegendre's,occurs ininvestigationsinapplied mathematics andnisusuallyanintegerthere;but,asinthecase oftheprecedingdifferentialequation,this limitation willnotbe imposedonthevalue ofn. 160 BESSEL'S[100 Tosolve theequation we-write- y=AjoTi+Aza?'*+Asxnt*+......... andsubstitute;wethenhave (m*-n')AX"1+V-9ia )AjKm'+(maa-rca )J-X'8+......... +AjOFf*+Ajf1**+.........=0, which must beidenticallysatisfied. Hence, from acomparisonol theindices, wehave mt=m i+2, wi,=ma+2, ortheseries isoneinascending powersofx,thecommon difference oftheindices ofthepowers being2;andthusmr=m1+2(r 1). Takingtheterm insowiththelowest index wehave m*=n~ sinceAlisnotzero;andtherefore m 1=+n,orm x= n. The coefficient of a)'"i+2r .ontheleft-hand sidemust bezero, andtherefore or,since m*=n*, A^-^>^i_-i"~~rtU 101. Consider firstthesolutioncorrespondingto 11^=+n. The coefficients Aarethengiven by AArA.,= 2V(w-t-r)' sothat forvalues ofrgreaterthanunity;andtheseries, which isa solution ofthedifferentialequation, becomes of '( 101.] EQUATION. 161 where A^isanarbitraryconstant. When toA1'isassigned the particularvalueonn.,where II(n)isGauss's function IIand is Zi11\ft>) thesame asT(n+T),then theexpressionisdenotedbyJn,so- that , =2 r-on(n+r)D(r) which isusuallycalled theBessel's function oforder n.When n ispositive, whetherintegralornot,theseriesproceedstoinfinity and, forfinite values ofthe variable,isobviously converging. ThusAJn,whereAisanarbitrary constant,isonesolution ofthe differentialequation.Beforeconsideringtheform ofJn,whenn isanegative integer,itisconvenient toobtain thesolution correspondingtothecase Thework isthesame asbefore with thechangeofsignofn, andthesolution is or1a* whereBtisanarbitraryconstant. To5Xassignthevalue =-,-r;then theresulting expressionisexactlythesame 2""II(-n)&r function of-nas/isof+nandmaytherefore bedenoted by J^,sothat . of.__1 2a(-n+l)"t"2I2X-n +l)(-?i+2) '"J (-1)' Ifnownbenegative,whetherintegralornot, or*fcepositive butnotintegral,this series proceedstoinfinity and, forfinite values ofthevariable,isconverging;inthiscase5J".Bisanother solution ofthedifferential equation. p.11 162 BESSEL'S[101. Ifthennbenotaninteger,whether itbeapositiveornegative quantity, Jnand/"_aretwoindependentanddeterminatepar- ticular solutions ofthedifferentialequationandtheprimitiveis 102. Ifnbeanintegerother than zero,twocases arise.First, ifnbeanegative integer andequaltop,azero factor occurs in thecoefficient ofallterms after x*pinclusive within thebracket ; andtherefore by86theterms whichprecedethisdisappear, and /..becomes or,what isthesamething, ,= sincen+p=0.Now this lastexpressionis(-l)p/pjthatis,ie (1)~V_ B;BOthat inthecasewhen nisanegative integerone oi theparticular solutions, Jn,degeneratesinto aconstantmultiple oftheother, 7_n. Similarlyitmaybeproved,oritmaybeatoncededuced from theforegoing,thatwhen nisapositive integer one ofthepar- ticular solutions, J_n)degeneratesintoaconstantmultipleofthe other,Jn. When niszero, thetwo solutions coincide. Hence ineverj casewhennisintegral whetherpositive, zero, ornegative, wemaj write butthat thisequation maybevalid itmust beremembered thai itrefers totherespective limitingforms oftheparticularsolutior ofthedifferentialequation when thesuperfluous terms ofthe latter forthespecial value ofnhave been removed from th< expressioninthegeneralcase;andtherelationmerely givesthi limiting forrn.Ithowever shews that.when nisintegraliti sufficient totakethepositive squarerootofw2andtoconsider, a thecorresponding particular solution, thefunction associated witl thatsquareroot. 102.] EQUATION. 163 Itthusremains tofindasecondparticularsolution intwo cases inorder tohave theprimitive;andthese twocases are First,when niszero : Second, whennisanintegerwhich (from theaboveexplanation) maybeconsideredpositive. 103.Toobtain theseparticular solutions itisconvenient tohavesome fundamental properties proved. Itmaybeatonce verified that (S) aiidfrom thelaattwowehave n^n n-1T *~dxJ=-x"Vn- Dividing the first ofthese throughout byj;71"1andthesecorfd byjr"""1 andsubtracting thelatter from theformer wehave Similarly 2 35 - tit Now itisevident from thegeneral value of/thatJv=0;hence thepre- ceding equations give Ex. Prove that _.this series isconverging. 4-...adinf.}. 164 BESSEL'S [104. 104.Toobtain thedesired1 particularsolution inthecasewhenniszero wesubstitute #=2*70+10 inthedifferential equation andtheresult is d?w 1dwTf&u ,Idu\ dudJ Tomake thecoefficient ofJvanish wehave d?u I ft fo?+vdx~' which issatisfied by theequation determining wisnow d*v> 1dw__2dJQ ifo*xdx~ xdx Nowfrom theequation d*Jn itfollows that istheParticular Integralof Thegeneralterm intheright-handsideoftheequation determining wis. xPn' wehave therefore forthisterm Hence andtherefore asolution oftheoriginal equationis } 104.] EQUATION. 165 Letthisbedenoted byT;thentheprimitiveoftheequation is y=AJ+BY, whereAandBarearbitraryconstants. 105.Toobtain thesecond particularsolution inthecasewhennisan integer wewrite y=J n\o%x-w, sothat d*w 1dw(.7? . T-S+-T-+I--T i}w=- Now da? a das Xbeing aconstant;andtherefore avalue ofwsatisfying is "-- Letwlbeaquantity satisfying da?^1efta^+^^ thenasuitable value ofwwillbe Theright-handsideoftheequation givingw:must betransformed. By thegeneralrelation between three successive Bessel's functions wehave - J-^1/0=1/2; hence 2(l)V1-2g hence also also 166 BESSEL'S[105. andsoon;andthegeneral equationis -," '''~ ufr^sfr"-~n-2~"' or,what isthesame equation, r/i-2- i Also,byaotual substitution wehave p+i-- +~xdx\ sothat,onwritingm=np }_ da . a?P ' p_-,p ^as+~p~ - - being aconstant. Ifpbenotzero, theright-hand side is ,^"" n;t-iI .o-/?-'^"-';;. /,-ftL' 'P ,p-p*lP-l* ^;u*^^A OT.,-->- while ifpbezero,theright-handside is tC***'":- i-si /?*-Ifnowwesubstitute intheequation forwlthevalue =-!,/ i-5>^--P' jj=0- ** acomparisonofthetwosides oftheequation gives ifpbenotzero,andgives ifpbezero;andtherefore, whatever pmay be, Hence thevalue ofwtis2-J 1n(n} 2 106.] EQUATION. 167 andtherefore thesecondparticularsolution ofBesseTsequationinthecasewhen nisapositive integerother than zero is i /n~PJ-in(n)2* Lettheright-handsidebedenoted byP"n;then theprimitiveisgivenby y=AJ n+3Y n. Ex. 1.Another method ofobtaining asecond particular solution isem- ployed byHankel asfollows. Anylinear function oftheparticularsolutions is alsoaparticular solution;hence inthegeneral casesuchasolution isgivenby which isthenperfectly determinate;while intheparticular case ofnan integerittakes theform 0/0since(-l)nJn=7_n.Prove thatwhen evaluated thisassumes theform "~J where *(a)=-rlogn() ; andidentifythiswiththesolution alreadyobtained. (Math. Ann. I.p.469.) Ex. 2.The series forJnisalways aconverging series; but,when zis large,theconvergenceisslowand itisconvenient tohaveaseries proceeding indescending powersofz.Prove that sothattheseries terminates,if2nbeequaltoanoddinteger. (Lommel.) 106. The relation between thetwoIpearly independentin- tegralsJnandJ_^inaybefound asin 96.Wehave do? 168 RELATION BETWEEN THEEQUATIONS OF[106. , d*JIdJ /_ n*\T811(1-T3* H---U^5+*-JK-= 5da? codoc \ a?j~" andtherefore whichgives _ do;~"dx us where .4isaconstant which, however, isnotarbitrarysinceJn and /_aredefinite functions. Toobtain thevalue ofAitis sufficient toconsider thehighesttermsonlyintheleft-hand side; when these aresubstituted, wefindthat ~n(n)n(-n) 2~n(n-1)n(-n) _2sinnir~* andtherefore nJ~^J= sinTITT, or,what isthesamething, d(J^\_2sinmr Ex. Obtain thecorresponding equation whennisaninteger. Relation between theequations ofLegendre andBessel. 107. Itispossibletoderive Bessel'sequation from that of Legendre. For,differentiatingtheequation mtimes, andwriting 107.] LEGENDRE AND BESSEL. 169 wehave jo-m . (1-^)^-(2m+2)a?^+ jre(n+1)-m(m+1)1z=0. Letthedependentvariable bechangedtowhere -(!-)*; theequation nowbecomes Lettheindependentvariable bechangedfrom asto <where 4>'=tt'(i-**); then afterslightreductions theequation becomes When wemakeninfinite, wehave which isBessel's differentialequation. When alltheseoperationsarecombined, wehave, astheresult, thatthelimit of when nisinfinite, isBessel's function ofordermt4>beingthe independentvariable. Itwouldappear from theforegoing process that</>isinfinite; thishowever isavoided bymakingecapproach indefinitely closely tothevalueunity. Thegeometrical analogueofthisrelation be- tween</>andxisthatwhereby anyverysmallportionofaspherical (orother) surface intheneighbourhoodofapointisstudied by assumingitultimatelytocoincide with thetangent planeofthe surface atthatpointandtobemagnifiedinthatplane. Ex.Verifythat theaboveexpression becomes,inthelimit, amultiple Inthisconnexion thestudent may consult Heine, Theorie derKugd- fimctionen, 2ndedition,vol.i.,p.182;Lord Rayleigh,Proc. Lond. Math. Soc. voL rx.p.61. 170 EICCATfS [107. Theprimitive ofBessel's differential equationhasbeen obtained forevery oaae;thefurther developmentofthepropertiesofthefunctions which occur inthatprimitive cannot begivenhere. Thestudent will findthefunctions fully treatedbyLommel inhisStudien uber dieBesseUsche Functioned andin severalpapers bythesame writer intheHatliematische Annalen,vols. n.m. iv.ix.srv. rvi.;inparticularthepaperinvoLxrv. deals with differential equations which areintegrable byBessel's functions. Reference should also bemade toNeumann's Theorie derBesseFschen Functionen andtoHeine's Theorie derKugelfwnctionen, 2nd edition, where(vol.I.p.189)alistof memoirsreferringtothefunctions isgiven;Todhunter's Functions ofLaplace, Lame"andBessel contains manyoftheproperties. Forageneral propertyofalllinear differential equationssimilar tothose which have justbeen discussed andwhichgiverisetofunctions depending uponaconstant parameterthestudent may consult, inaddition tothefore- going, Sturm, Liouville, vol. i.;andRouth, Proc. Land. Math. Soc. vol.x. RICCATI'SEquation. 108. Riccati's differentialequationis +v-! but itisconvenient toconsider firstthemoregeneralform as -^-ay+ly*=caF. Ifinthelatter theindependentvariable bechangedfrom SBto z,where z=xa ,andthedependent bechangedfromytou,where y=uz}theequationbecomes du,.bac--*+-u!t=-za ,dza a which isRiccati's form. 109. Consider nowthemoregeneralform. Firstly,itcanbeintegrated infinitetermswhenn=2a. Forassuming y=us?wefindonsubstitution aT1^+baFu*=cat1 , doc sothat a1"*^+M=ca?-*a . das Inthecasewhen n=2athisbecomes .._du 109.] EQUATION. 171 thevariables areseparable anduisexpressibleinterms ofex- ponential,orcircular, functionsaccordingasband cLave, orhave not,likesigns. Secondly,itcanbeintegrated infinite termswhen(n2a)/Znis apositive integer. Letthedependentvariable bechanged fromytoyvwhere ccn A-\=yandAisaconstant thevalue ofwhich hasvettobe 2/i_ _J determined. When substitution takesplace andtheterms are rearranged,theequation becomes WechooseAsothattheconstant term vanishes, andthusA= ora/6. Takingthevalue a/6forAandsubstitutinginthisnewform wehave, afteraslight change, a -jj-(a+n)y1+cy*=boc". Now thisequationisofthesameform asthatwithwhich we began;andthechanges,thathave takenplace, areinthe coeffi- cients theoriginalahaschangedtoa+n,aud 6and chave changed places.Inthis lastequation wewrite a+n ccn *--T^ theforegoing analysisthenshews thattheequationinyawillbe Andtheresult ofisuccessive transformations willbetoreduce the given- equationeither to tc-^ (a+in)yt+cyf=bof1 orto accordingasiisoddoreven. 172 RIOCATI'S [109. Now,bythecase firstconsidered, thisequationisintegrablein finite terms, if n=2(a4-MI), ,,,.. n-2a that is.if s- 2n isapositive integer. Takingnext thevalue zero forAwecaneasilytransform the equationinto anequationwhich differs from theformer iny^onlysofaras regardsthesignofa.Adopting now forthistheprecedingseries oftransformations wewrite nax andtheequationinyais x as?" Hence after i1transformations ofthis series (andtherefore after itransformations inall)thegiven equationisreduced either to orto xj~(~a)y<+^'8~aB"' Ineither casetheequationisintegrableinfinite terms, if 7i=2(in a), ,. . . n+2athatis,if s 2ft isapositive integer. Combiningthen thesetworesults wehave :theequation^ as~ay+by*=casn isintegrable infinite termswhen (n2a)j2nisapositive integer. Ineach case theintegralisgivenintheform ofafinite continued fraction, the lastdenominator ofwhich involves either exponentialorcircular functions. 174 RELATION BETWEEN THEEQUATIONS OF[111. someimportant transformations which render them linear ofthe second order. InE-iccati'sequationletthedependentvariable "bechanged fromutovwhere , 1dvou=-j-,vdx sothat ifuisexpressibleinfinite terms, vwillbesoalso;the equationthenbecomes whichmightbetaken asastandard form, equivalenttoRiccati's equation. Ifband chave thesamesign (inwhich caseexponentialfunc- tions occur in)thisequation maybewritten while iftheirsignsbeunlike(inwhich case circular functions occur inu)theequationis Both ofthese areintegrableinafinite form forthesame value of mthatrenders Biccati'sequation integrable. Changetheindependentvariable fromxtoz,where .qz=a? and q=km+1=-say ; i j > theequationthenbecomesn d'vnIdv , . -j-j--j bcv=0. dsr zdz This therefore isintegrableinafinite form if 1i T -,M 1 n whence itfollows thatw.must beequaltoanoddinteger;andso iftheequation bewritten cZau2pdv,jj-*- -j-JCOCV=0,dsr zdz 111.] BESSEL AND BICCATI. 175 thecondition ofintegrabilityinafinite form isthatpshould be aninteger. This isreducible toitsnormal formbythesubstitution andtheequationforwisvz~*=w, which isintegrableinafinite form ifpbeaninteger. Lastly,letw=z^tbesubstitutedjtheequation fortis d*t 1dt, 1Nat . S'+iS-^-U'+tfp-' theprimitiveofwhich is *=AJf+i {(-be?}+JJ-cp+D [z(-left. Ifp+%beaninteger,thisceases tobetheprimitive; wethen have fortheprimitive *=AJm [z(- be)*}+B7p+it[z(-bcfy Hence thesolution ofRiccati'sequation canbeeaypressedinterms ofBessel'sfunctions ;and,inparticular,theprimitive of isgiven by or __m+2 m+2 accordingasm+2isnot,oris,thereciprocal ofaninteger. This isimmediatelyderivable from acombination ofthe precedingtransformations. Theonlycaseoffailure isthat inwhichm+2iszero, thatis, when wz,is-2;theequationisthen+\va>n= dao =o*AJj_(z\)+BJ__i_m+2 w+2 which canbesolved bythemethod of47. 110.] EQUATION. 173 110.Wecannowobtain conditions that Rlccati'sequation shallbeintegrableinfinite terms. From 108 itfollows that istransformed bythesubstitution u=y/a;into *%,-*+&=<*, where ra=n 2.Now thelatterequationissointegrablewhen n2=2m, where iisapositive integer ;andtherefore Biccati'sequationis integrableinfinite terms if m+22=2i'(m+2). Takingthenegative signwehave 4im=-27=T; while thepositive signgives orwhat istheBarnethinginthecase ofthelatter bymerely changingtheintegeri. Hence Ricoati'sequationisintegrableinfinite terms, if m= zTTi' ibeingzeroorapositive integer. Ex.Prove thattheequation cLiL dx~~ isintegrableiufinite terms,if *m+l -2i+l -2i-l or =-; =, ibeing aninteger. Relationbetween ikeequations ofBessel andRicoati. 111.Theequationsof108intheform inwhich theyhave been discussed areofthefirst order, butarenotlinear;there are 176 SYMBOLICAL[111. Forfurther information upon+hinequation amemoir byJ.W.L.Glaisher inthePhil. Trans. 1881, pp.759 828,should beconsulted, where full references toauthorities willbefound;andtheconnexion between Riccati's equation andBessel's willbefoundfully discussed inthebookandpapersof Lommel towhich reference hasalready (p.170)beenmade. Some examplesofthesolutionexpressed byseries willbefound inthe Miscellaneous Examples. SymbolicalSolutions. 112. Incaseswhen thesolution ofadifferentialequationin series consists ofafunction inafinite form orwhen itconsists of aterminatingseriestogetherwithsome function orfunctions ina finite form, itissometimespossibletoobtain asolution ofa symbolicalnature which will,when theoperations therein indicated areperformed, prove equivalenttothesolution otherwise obtained. Asanexample,consider thedifferentialequation d8 ?/ =m(ra+1)__U.__m'-tlx_ ini dx*nyofy' thesolution ofwhich hasbeenprovedtobeexpressible inafinite formwhenmisaninteger. When thedependent variable is transformed fromytoubymeans oftherelation y=uasn+1 , theequationbecomes .1du, Consider nowthedifferentialequation thegeneral integralofwhich is andchangetheindependentvariable fromxtoz,where zstands for\o? ;theequationbecomes .+*rt-O. dsr dz Let thisbedifferentiated m+1times withregardtozand dn+1v let tdenote T-SL 5thenwehave * 112.] SOLUTIONS. 177 Letnowtheindependent variable berechaneed from zto cs; * O' theequationthenbecomes , dor CB da: Hence wehave u=t ,-\/7\ -Sal) theprimitiveoftheoriginal equationinytherefore is (1rf\m*1 1|)<^+ao Aslightlydifferent formmaybegiventothis, for onchangingthearbitrary constants; andtheprimitive maybe written intheform x Since thedifferential equationremains unaltered, when form issubstituted (m+1),theprimitive maybeexpressedinthe additional forms Ex. 1.From theforegoingitcanbeatoncededuced thattheprimi- tiveof 6 (anequation arisingininvestigationsconnected withtheFigure"oftheEarth) isexpressibleintheform F.sn 12 178 SYMBOLICAL SOLUTIONS. [112. Ex. 2.Prove thattheprimitiveofthedifferentialequation d can,inthecasewhenqisthereciprocalofanoddintegral 2i+ 1,beexhibited intheforms (Glaisher.) .Sfc. 3.Prove thattheprimitiveoftheequation oPu , p(p+I) -j-a+a2tt=t-^-sLudor aP isgivenby. where ristobeputequal toa2aftertheperformance ofthedifferentiations. (Gaskin.) Inallthese caseswhen thesolution oftheequationisthusgiven symboli- cally,itisnotdifficult toidentify thesolution inthisformwiththatobtained inanyother form, such asoneinseriesbytheearlier methods ofthischapter, orasonebymeans ofdefiniteintegrals asindicated inchapter vn.The student whowishes forfuller information onthesubjectofthesesymbolical solutions and their connexion with solutions inother forms willfindafull discussion inthememoir(Section vi.)byJ.W.L.Glaisheralready (p.176) quoted. mSOELLANEOUS EXAMPLES. 1.Integrateinseries, andexpressina-finite form theintegrals of,the equations andintegrate MISCELLANEOUS EXAMPLES. 2.Solve theequations'--<179 (iv) 3.Integrateinseries thedifferentialequation andexpress theintegralinthefinite form A{1-(1- 4.Verify thatarootoftheequation satisfies(Glaisher.) (Spitzer.) 5.Transform theequation byassuming y=eaaandm+ai+na=f ( )*,into integratethelastequationinseries. 6.Obtain theprimitiveoftheequation &y ,<fyty da?^~qdx= a? intheform qssy=A(qx-2)+B($x+2)e-*. 7.Obtain theprimitiveoftheequation intheform (LesUe Ellis.) 122 180 MISCELLANEOUS 8.Prove thatthecoefficient ofcf1intheexpansioninascending powers ofaof (l isasolution of 9.Prove that, withthenotation used forthesolution ofLegendre's equa- tion,{Pn(cos0)}3isasolution ofthedifferential equation ' 10.Prove that,with thenotation of90,91, (Trinity Fellowship Examination, 1884.) 11.Prove thattheprimitiveoftheequation isgivenby provided mbenotgreater than n. What istheprimitive whenmisgreater thann9 (Heine.) 12.Shew thatthesolution oftheequation where kisaninteger, maybeexpressedintheform whereynisthesolution ofLegendre's equation. 13.Obtain theprimitive oftheequation (Heine.)' 14.Prove thattheequation has,inthecasewhennisaninteger, forits'primitive }}. (LornmeL)- EXAMPLES. 181 15.Obtain theprimitive oftheequation intheform where ;%'- (n-!)'-&. (Lommel) 16.Verifythattheprimitiveof is y-a 2 33=0 wherea,,,t^,..., Om-jaretheroots oftheequation am=l;andthatof a y where a,a!,...,a^aretheroots ofaaiB+1=-i, (LommeL) 17.Theprimitive oftheequation is y=A andthat of #*^-4 i i is y=x{AJ (e*)+Y (e'i )}. (Lommel.) (See,forconnexion between thesetwoequations, Ex.10,p.127.) 18.Prove that, withthenotation of101, "A"TfX nnotbeinganinteger, andthat -TJ=-l n+l= a.1 (Lommel.) 19.The differentialequation isintegrableinfinite terms, whatever function ofa;isdenoted byQ,provided mbeaninteger. 182 MISCELLANEOUS 20.Theequation isintegrableinfinite terms,if 2{(l-r)a+4c}i= ' where iisapositive integerorzero. (Malmsten.) 21.Prove that thecoefficient ofh?*1intheexpansion ofa(yS>+ash) satisfies thedifferential equation (Glaisher.) 22.Shew that,ify=Xbeasolution oftheequation (kbeingaconstant), thenasolution of isgiven by v. Hence solve theequation cPy 4dy ,dS-52+**-- (Leslie Ellis.) 23.Theequation a-^3--J--o isintegr^bleinfinite terms inthefollowing cases : 1.when -isanoddinteger ; ct/ (/j\a c-j 2.when-^fl J+4-Visanoddinteger ; 3.when--|(l )+4-[isanoddinteger.a(\ a.) a] 24.Prov thattheequation_ (a admits offinite solution, EXAMPLES. 183 1.whenanyoneofthefourquantities a- /8isaneven integer, 2.whenanytwoofthequantities areoddintegers;where a1903andj8ls/32aretheroots oftherespective quadratic equations (a-2)(na-2n-2)+%m(a-2)+g='0, aud Janj3 (?i|3-2)+$<i+/=0. (Pfaff.) 25.Prove thatthethreeexpressions 1a2^21 dJ \P areallparticularsolutions oftheequation andshew that,whenpisnotaninteger,these threeexpressionsareequalto one another. Obtain,inthis case, asecond andindependent particular solution. 26.Prove thattheprimitiveof -r do?a maybewritten ineither oftheforms (/7\p+l a?|jj{^- (Boole.) 'Prove thattheprimitiveofthesame equation mayalsobewritten inthe form (Donkin.) 184 MISCELLANEOUS EXAMPLES. 27.Theprimitiveoftheequation canbeexpressedintheform Obtain that of intheformd*y dy-^-+my=x-f-ds?adss (Spitzer.) 28.Theorthogonal trajectoryofthesystemofsurfaces ofrevolution givenbyPn=crn+1 ,wherePnisthesolution ofLegendre's equation and its argumenta?isthecosine ofthevectorial angle ofanypoint,isgiven bythe equation 29.Prove that,iftheequation betransformed bytherelations s(ca?+cZ) =a0+6and#=(ca7+ctf)asothatu isthenewdependent variable, thenewequationis where Hence, orotherwise, solve theequation cPy y da* CHAPTEE VI. HYPEEGEOMETBIC SERIES. 113.THE series a/3 a(a+1)008+1 1.2.7(7+!) 1 .2 .O.7 iscalled thehypergeometricseries and isusually denoted byF(a,ft,7,as) ;thefourquantities a,ft,7,xarecalled itselements andofthese soalone isvariable. Theelements aand ftmaybe interchangedwithoutaffectingthevalue ofF;ifeither ofthem beanegative integertheseries will consist ofafinitenumber of terms, otherwise itwillproceedtoinfinity.Itwillbeassumed that7isnotanegative integer,sothat infinite termsmaybe excluded. Ifsobelessthan 1,theseries isconverging; but ifa;begreater than1,the series isdiverging.Ifsobeunity,theseries iscon- vergingif7aftbepositive, anddivergingif7aftbezero ornegative. The series isone ofvery great generality andincludes as particular examples verymanyofthe series which occur in analysis. Thefollowing examplesadmit ofeasyverification : H(1+oOB+(1-)'=2F(-fa-$n+iia?). m.log(I+at)'=<cF(l, 1,2,- as). 186 DIFFERENTIAL EQUATION OFTHE[113. IV.log V. /gP\ VI. coshx=F(a,@, ,-j-~],whena=oo=. VII. cosnx=Ftyn, fan, ,sin9 a;). jfik. 1.Prove that allthedifferential coefficients oftheseries willbe divergingforthevalue#=1 iftheaeries itself bedivergingforthat value; audthat allthedifferential coefficients fromandafter oneofsome order will bedivergingforthevaluex=Ithough theseries beconvergingforthatvalue. Ex. 2.Expressashvpergeometricseries (i)sint,thevariable element intheseries beingZ2 ; (ii) saint, thevariable element intheseries beingsin2*; (iii)ocant,thevariable element intheseriesbeing-tana . Others aregiven byGauss atthebeginningofhisearlier memoir(referred toin 134). 114 Letthecoefficient ofofbewrittenAr;thentherelation connectingconsecutive A'B is Consider thedifferential equation =...............(i) inwhich S-stands fortheoperatorx-5-.Asolution ofthisequa- tioncanbeobtained inaseries :letthis series begivenby Substitute thisvalue inthedifferential equation, which must beidenticallysatisfied;each separate powerofasmust therefore disappearinvirtue ofthequantity multiplyingitbeingzero. Thus forthelowest power wehave andfrom thevanishingofthecoefficients ofthehigher powersthe relation between thesuccessivequantities Bisgivenby 114.] HYPERGEOMETRIC SERIES. 187 We shallassume thatBQisnotzero,because therelationB= would make alltheJ?Bzero; andthus theformerequationis satisfiedbyeither fL=Q Or fj,=17. 115. Take firstthevalue/-t=0;thentherelation connecting thequantities Bbecomes NowwhenBQ=1=A,therelationjustproved, comparedwith thatwhich connects theA'e,shews thatBr=Ar;andtherefore the series assumed forybecomes thehypergeometricseries. Thus onesolution ofthedifferentialequation (i)isF(a, 13,%<a). Lettheoperatingfactors in(i)beexpanded andterms ofthe same order collected;then theequation maybewritten cva B ,^=x-J-I+BT; ewo dan when these values areinserted theaboveequation,afterrearrange- mentanddivision byof(1 as),becomes c^aa(l-) da(l-)".........) which isthedifferentialequationsatisfied by.F(a, fi,7,no). Take next thevalue/i=17 ;therelationconnectingthe quantities Bbecomes LetS=I;thisequation shews thatthequantities Barethe successive coefficients inahypergeometricseries whose constant elements arerespectivelya+17,#+17,27.The series assumed forybeginswith acl~y ;hence thevalue ofyis at-vF(a+I -7,/3+1-7,2-7,aj), andthisalso isasolution ofthedifferential equation (1). 188 NOEMAL FORM OFTHEEQUATION. [115. Wehave thustwoparticularsolutions ofthis differential equation ;andtherefore anyotherparticularsolution which is finite forvalues ofxlessthanunitymayberepresented "by AF(a t13,ry,as)+B&-*F(*+1-y,+I-y,2-7,a), inwhichAandBareconstants, thevalues ofwhich maybe determined bycomparing powersofx.Ifinthisexpression A andBdenotearbitrary constants, itfurnishes theprimitiveof(1). 116.Toreduce(1)toitsnormal formwemustcompareit with thegenerallinear equationofthesecond order.Wethen have _ ss(1x)to 1x andtherefore theinvariant /,being becomes, aftersome reductions, i-x I-*,8' v- where Letthisinvariant bedenoted either byIori/r(x) ;the latter form willbeconvenient when theindependentvariable comes to bechanged. Thusequation (1),bythesubstitution becomes ____|_y^r(#)=o , (2), inwhichty(as)denotes theforegoingfunction of . 117.]SETOFPARTICULAR SOLUTIONS. 189 Setof24particularsolutions. 117.Wenowproceedtofindsome furtherparticular solutions ofthis differential equation.Itfollows from theinvestigation of64thattheconditions, which mustbesatisfied inorder that theequations d*v and -g+**i(*)= ........................(3) should betransformable intooneanother are, firstly, /dn-* "-*UJ=zu> andsecondly, Hence, ifweconsider^()asagivenfunction oft,thelatter equationwillgivethevalue oftinterms -of;andwhen this value is-fdund theformer will furnish the relation between vand z. Nowassume that thefunction^ (tf)issuch astomake equation (3)thenormal form oftheequationsatisfied by ahypergeometricseries with constant elements a7 ,ft',7' ;and supposethatwecanobtain from(4)avalue oftinterms ofx~ Then, since thevalue ofuwillbeatoncederivable from that oft, wehave asolution of(2)intheform vt (1-QW*1-F(a', ft',,/,*) ; andthis isdistinct from thevalue ofvwhichwehavealreadyhad. 118.Theprimitiveof(4)willgivetasafunction ofec,a,ft,y,. a'ifi>7' ;I*usselect those forms ofthisfunction which make t dependenton atalone, andindependentofthetwo setsofconstant. elements. Wemay,toobtain these, write *,a=0, 190 PARTICULAR[118. Theformer ofthese onmultiplication byi'~* isdirectlyinte- grateintheform andproceedingwiththeintegration,wehave A4 C(Cx+C') _aas+b cx+d onchangingtheconstants. This isthegeneralvalue oftwhich makes thefunction{t,as]vanish;buttheconditionsrequirethat (ad-&c)a ,feus+b\Or-Tilrt-j=Yl\ca:+dJ andthiswillnotbesatisfied forarbitrary values ofthese constants, which must therefore bedetermined soastobeindependentof theconstant elements oftheseries. Now where A=I/u,", andwemaywrite ,... ,^(*)=i Hence theconstantsa,b,c,dmust besuch astosatisfy =(ad-6Ha { } Thequantities a,@,y(and therefore A,B,Gwhich arefunctions ofthem(are arbitrary andthus thenumerator anddenominator oftheleft-hand fraction canhave nocommon factorexcepta 118.]SOLUTIONS. constant; and similarlyfortheright-hand aide. Hence wemay write m(Ax9 =(ad-6c)a [A'(ax+&)"+B'(ax+b)(ca+d)+0'(ecu+df], mx* (1-fl)2=(a*+by(ca+df{(c-a)x +d-6}", inwhichmisconstant. The latter oftheseequationswilldeter- mine thevalues ofa,b,c,d.which areadmissible;theformer will then serve toindicate therelations ofof,{?,7'toa,ft,7inorder thattheexpressionattheendof117maybeasolution of(1). 119. Comparing nowthecoefficients ofthedifferentpowers of CDonthetwo sides ofthe latterequation, wefindthatthe followingsets ofvalues fortheconstants willmake theequation identicallysatisfied : (i)c==6=ad;m=a; (ii)c=Q=d-b =a+b;m=a9 ; (iii)a==d=c &;m=68 ; (iv)a==d-b=c+d;m=be ; (v)&==ca=c+d;m=ae ', (vi)d==ca=a+b;m=b9 . These values substitutedsuccessivelyintheexpression for tin terms ofasgive: (i)t=co; (ii)t=1x; (iii)t=- ;x respectively;andthese form thecomplete systemofvalues oft required 120.Wenow transform the first ofthetwoequations by means ofeach ofthese inturnandobtain thenecessaryrelations between a',fB',7'and a,$,7. Consider firstthesetofvalues(i).Wehave sothat 192 PABTIOULAB[120. or,what isanequivalentsetofequations, Whenexpressedinterms oftheconstant elements, these relations are and(rememberingthataninterchangeofthe firstandsecond con- stant elements makes nochangeinahypergeometric series), we findthatthese aresatisfied by (1)rf-a .................. P=P.................. 7=7; (2)a'=7-a ............/3'=7-............ 7=7J (3)a'=a-7+l.........#=-7+1......7'=2- 7 ; (4)a'=l-a ............,<3'=l-/9 ............ 7'=2-7. Since t= cc,-^-isunityandtherefore uisunityforthisvalue oft;andtheparticularsolutions ofthevequation, which cor- respondtothese four setsofvalues, arerespectively -a,7-7j , _7+1,^_7+l,2-7,a;), aj1-*7 (1- )w>^(1-a,1-A2-7,a). Now these aresolutions ofequation (2) ;inorder toobtain the correspondingsolutions ofequation (1)wemustmultiplyeach of themby andtherefore fourparticularsolutions ofequation (1)are (I) y=*<,A7f); (II) y=(1-)*' J'Oy-q, 7-&7,x) ; (01) y (IV) 2/ 120.] SOLUTIONS. 193 Treating nowtherelation t=1 asinthesameway,wefind other fourparticularsolutions intheforms (V) y-J'(alAo (VI)y=-yF(a-7+!,-? +!,a+-y+l,1-a); (VII) y=(l-fl;ra Andfrom therelation t=-wehave asoneparticular solution (IX) y=x-* 121. Alltheparticularsolutions forthedifferent values oft canbefound intheabove manner. Each value oftleads tofour particular solutions, sothatthere areinall24ofthese. But this laborious method ofobtainingtheremainder need notnowbe adopted ;itispossibletowrite down, from thenineforegoing,the followingfifteen tocompletetheset : (X) y=a,^ (XI) y=*?-< (XII) y-^- (XV) y=-y(\-& (XVI) y=oc--y(\-a (XVII) y=(l-x)"L . F. 13 194 RELATIONS BETWEEN THE [121. (XIX) y-^-^ (XX) 2/=^1-y (XXI) y-<T' (XXII) yssflr' . (XXLH) y=xa~>(I- (XXTV) y=flJp"Y(l-aj)Y~a" fl7fl-0l7-0,7-a-n -^al Relations between theparticularsolutions. 122. Let allthese solutions bedenotedby thesuffixes andthenumbers oftheforegoing equations correspond- ingtooneanother;thesequantities yarenotindependent, for,by theordinary propertyofalinear differentialequationofthesecond order(ofwhichtheyallaresolutions),there isbetweenanythree ofthem x, .>*arelation oftheform andwemust findthese relations forthedifferent combinations of thesolutions. Butcertain cases will arise inwhich eitherAorB willbezero,andtherefore thecorrespondingsolutions will differ from oneanotheronlybyaconstant factor;andthese canbe recognised bytheapplicationofthefollowinglemma. Ifthere betwosolutionsofthedifferential equation (1)developed inthesameascending powers ofCDandbothseries beconverging,then they differ from,oneanotheronly byaconstantfactor. 122.] PAETICULAJR SOLUTIONS. 195 Forthesake ofsimplicity supposeoneofthesolutions tobe F(a,ft,7,cc}andtheotherwhendevelopedinascending powers ofxtobegiven by y=A+Bx+Oa?+......... Substitutingthisvalue ofyinthedifferentialequation weshould, lojaprocesssimilar tothatin114,findy=AF(z, fi,y,as),which provesthelemma. 123. Letusapplythislemma toobtain theparticularso- lutions which areequaltoyl;thisweshallsupposetobeacon- verging series, sothat as<1.Then yaisalsoaconvergingseries proceedinginthesameascending powersofasasyl;the firstterm ineach isunity;theconstant factor ofthelemma istherefore 1andwehave #1= y,- Thenextoneinthe listwhich, expandedinascending powersofso, beginswith 00isys;ifweselect from thecoefficient ofxn ,weshall find ittobe 1.2...... JF'(a+n>/9+n,a+/9-y+n+l,1). But iathis coefficient Fisconverging (andsohasafinitevalue) onlyif bepositive (see 113), that is,if17nbepositive. Hence fromand aftersome definite term thecoefficients ofthepowers of CDwillbedivergingseries;andwecannot then consider the aeriesF(a,/3,a+/9-7+1,1cc)tobeconverging thoughex- pansibleinascending powersofx.Hence y6isnotequaltoyv DealingwithyltyiVywya,ywinthesamewayitwillbe found thatthe lasttwoalone areconvergingseries atthesame time asF(a,fi,y,as) ;andhencewehave yttajadyvytandyvylQandyv,ymandy1&arederived from each other byexactlysimilar transformations ofelements ;132 196 THEPARTICULAR SOLUTIONS.[123. thus topassfromy^toystheformer ismultiplied bya^-y, thenew firstandsecond elements"beingobtained bysubtractingtheold third from theoldfirstandsecond andadding unitytoeachresult, andthenew third element bysubtracting theoldthird element from 2.Thisprocessisseen tobethesame forallandtherefore 2/8= 2/4= 2/19= 2/20.....................(ii). Ex. Prove that #6=Vo=yu=yn........................... P), (v), 124. Itthusappearsthatthe24solutions canbedivided into sixclasses;andtheequalmembers ofthese classes wemaydenote respectively byFt,Fa,FB,F4,FB,Fecorrespondingtotheabove sets ofquantitiesinorder. Itremains tofindsuch relations aa theremaybebetween theseowingtothefactthattheyaresolu- tions ofthedifferentialequation. NowFnandY4areconvergingforthose values ofccwhich are lessthan 1,whileF5andF6areconvergingforthose values ofor which aregreaterthan 1;astheformer therefore areconverging- while thelatter arediverging and vice versa, there canevidently benoequations connecting FBandF4withFBandFa.Wethere- foremust find theequations betweenanythree ofthe set YltTvTvF4;andanythree ofthesetFz,Fa,FB,F fl;and itwill besufficient tohave thoseequationsintowhich7lenters, as,by changesoftheelements and division byafactorthroughout, any one ofthe'quantities Fcould betransformed intoF,.Thus theequations requiredwillbethoseconnectingthefollowingsix groups: FFFFFF-FFF-FFF-F VV *v*v^8'Ji'*a*4>*vJa>-*4>* i>-*a>*G>-*i>*v**> FFF *1*6>*6' Lettheequationforthe firat ofthese groupsbe or Todetermine MandNthesubstitution ofanytwoparticular values ofacwillbesufficient;letthen as=1andx=0,andsuppose 124.] GAUSS'S nFUNCTION. 197 17apositive quantitysothat tf1"7iszerowhen as= ;wehave forthesetwocases ToevaluateMandNwemust obtain therelations hetween theseries forargument unity,towhich wenowproceed. IntroductionofGaws's IIfunction. 125.The coefficient ofxmin IS a(+l) 1.2 ......... 771.7(7 +!)......(7+771-1) 7-1 7(7-1)'1.2. 3......(ra-1).(7+1)...(7+m-l) =coefficient ofoomin--, 1NJ1 (a+1,/3+1,7+1,a) ;7(.7- -1-; andtheterm ontheleft-hand sideindependentofecvanishes so that 71 Butfrom thedifferentialequationsatisfied byF(a,0,7,a;)we have Letthevalue ofJ7 (a, /9,7,x)when#ismadeunity bedenoted d*F byFl(a,/3,7);thevalue of-pjwhen a;ismadeunityisfiniteand CUB therefore 198GAUSS'S *ifrA7)-*;(,&7-1)---~ T- _ aft (7-l)(y-a-/3~l) BOthatFQ_ 1-(fy- (7-J.)(7--/9-l) or,changing 7into7+13wehave ' Similarly y A/g|Jfl 1 andtherefore 126. Let 1.2.3 ~ov( 7<k*bedenoted byII(kz\-&) (JS+K)J \>I' then Since wehave =1.2. 3...*.(*-+ 126.] nFUNCTION. 199 andso 1.2. 3......e onthesuppositionthat zisaninteger. From thistransformation andfromtheoriginaldefinition alikewehave Theseequations shew that foragivenvalue ofzthefunction II(k,z)tends towards alimitingvalue askapproaches infinity, andthatthislimiting value isfinite. Asthen II(oo,z)isafunction ofzalone, letithedenoted byII(z}\thelastequation shews that H(z+1)=(g+1)H(z), andtheformer shews that, ifzbeaninteger, n(*)=*i, while inanycasewehave n<=r(*+i), where T(*+ 1)istheGamma Function ofEuler. Intheequation givingF1letkbecome infinite; thenevery term oftheseriesF t(a,ft,y+oo;iszeroexcept thefirst,which is unity.Ifwesubstitute forII(oo,71)andtheother functions their values II(7 1),wehave Ex. 1.From theexpansion of tinaseries ofascending powersofsint, provethat Ex. 2.Prove that n(-e)U.(e-1)=TTooseo en. Ex. 3.Obtain therelations (i)F1(a,l-t,y)F 1(-a,l3,y-a')=l; (ii) Fj.(a,fty)F i(a,-/9,y-/9)=l. 200 RELATIONS BETWEEN THE [126. Ex. 4.Prove that n^n^n(z- 1)n (i-1).........n(,-^)=(27r)*f"^n(^). (Gauas.) Determination ofconstants intherelations of124. 127.Theequationsof124nowbecome _nQ3- 7)ii(- 7) andtherefore n(-a)n(-/9) fromwhich withtheuseofExample2intheprecedingset itis not difficult todeduce*that n(7-l)n(a-ry)IT(/3- 7) These then arethevalues oftheconstants intheequation (i)Y^MY. Similarly,ifwewrite (ii)T^M. wefindthatthevalues ofM^andN^are ,T_n(-a)n(-<8)11n(7a-/9)n(- 7)- Itiseasytoshew thatthefollowingarethefourequations correspondingtotheother fourgroupsinorder: (iii)Y 127.] PARTICULAR SOLUTIONS. 201 i where^=n(7-1)n(7-a-1) 9 AT_n(- (iv)Y^MJ^ ,n- wheren(1__^u~(^lju (y-ft-I)' !!(<*-) II(-7)' (v)Y^MJ^NJ, where jf-n(y-i)ii(/?- 7)n(-a)whereM- where^.?(7-l)n(^-a-1)wneroM~ (7-a-1)' Itshould beremarked thatthelabour ofdeducingthese con- stants neednotberepeatedforeachequation;eachequationwith itsconstants canbededuced from the firstequationand itscon- stants. 128.Wenowpasstoadifferent setofequations which connect anytwooftheparticularsolutions andtheir differential coefficients. Ithasbeenproved that, ifY 1andFabetwoparticularsolutions oftheequation whereGhasaconstant value which depends uponthepairof particularsolutions selected. Inthecasewhen theequationis that satisfied bythehypergeometricserieswehave 202 RELATIONS BETWEEN THE[128. ft=77-a-/3-l a;(1 as)x I os. andtherefore Thevalue of(7inanyequation maybedetermined either bya comparisonofcoefficients ofthesamepowerofwonthetwosides orbythesubstitution ofaparticularvalue of as. Example1.Let Leteach sidebeexpandedinascending powersofx;theterm involvingthelowestpowerofsoin Yl~daJ aR is <cl~y ;theterminvolvingthelowest powerofa;in ~FS~des is(17)oTy ;henceequatingthe coefficients ofthelowest powers wehave andtherefore 2/i ^2/B f -t \iO.-B iynr- y,-5=(y 1)#(1 os)~~ JadosJldasw ' v ' Example2.Let Weproved before that inwhichMand ^7"aredefinite constants. Thisgivesondifferen- tiation da das das' 128.] PABTICULAE SOLUTIONS. 203 andtherefore from theresult ofthelastexample. Nowfrom thevalues ofM andNwehave M_n(7-1)n(-7)n(a+-7) ~ Butn andtherefore andtheequationbecomes Vtyl_tys_n(7-l)II _ 2/8dxy*dx~ n(a-l)*U ' Ex.Prove that 7~y(1 tf)V~a andthat 129. Inalltheforegoing investigationsthequantities a,@,7 havebeensupposedtobeindependent,andtheseries have con- sequentlyretained their moatgeneralform;butmany important applicationsaremade byassigningeither oneortworelations between thethree constant elements, orbygivingnumerical values tooneormore ofthem. Suchapplications (asforinstance toelliptic integrals)cannot bediscussed here;butthestudent whowishes forinformation onthesepointswill find attheendof thechapteralistofthemoreimportantmemoirsdealingwith hypergeometricseries. 204 OASES OFSOLUTION[130. Specialcasesofintegrationinafinite form. 130.Wepassnow toconsider somespecialcaseswhen the hypergeometricseries canbeexpressedinafinite form. Ithasbeen proved (61)that thequotientsofanytwopar- ticular solutions oftheequation satisfies theequation I{s,*}=I, where/isafunction ofxonly ;and ithasbeen further shewn that, fromanyparticularvalue ofswhich satisfies thisequation, thevalue ofthetwoparticularsolutions oftheformerequation canbeobtained. Inthecase ofthehypergeometricseries the value of/is !-' x'-g'+F"-!-!A _ o-i)9(*-i) J.........w> \,ft,vbeingdefinite functions oftheconstants a,ftand7;sothat forthis series the differentialequationwhichgivessmaybe written Ifthen arelation between 8and a?canbefound which is expressibleinfinite terms, itfollows from theformulas of62that thehypergeometricseries willbeexpressibleinfinite terms. This cannot beexpectedtooccur inthecasewhen theparametersare general;from thefewinstancesgivenitwillbeseen that the values of\ft,varedefinite numerical constants. There areinallfifteenseparate cases, andnomore;forthe proofofthis,reference should bemade inthe firstplacetothe memoirs ofSchwarz (see 134) towhom theinvestigation,ina completelydifferent form, isoriginallydue. Itisconvenient torecapitulate herethegeneralfonnulee oftransfoimation ofthefunction{a,x}forthechangesofthevariables'; thespecial examples giveninEx.3,62areparticularcases ofthegeneralrelations which are INAFINITE FORM. fiitf-f-fcI JJ.In=..... I'M'-H/' J mlilitiniiuli-.\.uii]ili-swumuytuku205 (ii). r.irniiilih wlih-h AvillIIIHIVO iiHuful,\nthutwhich uri t-;llu-iiwnImvo...<iv). byan thillt-' ami,a HIItlmt\a,.&} wliii-hmuyIHIwrittfii inoithor ufthoforma .(V). 1H1. CAHE I. BywritingA"=xin(i)intho fonrraluujustunumoratod wohavu byaHoriflH ofjiroperHubstitutionH womaypassfrom this tothecorresponding equationforthehypergcomctricsuriua. 206 CASES OFSOLUTION [131. Firstly,let <r then{s,a?}={S,a]+ (^ while by(iii) Buto-=5n ;therefore andthus Secondly,let y=5a=l- sothattherelation between sand#is +1,then Again using (i),wehave {s,}={r, butinthis dT_~ ' sothatwehave 131.]INAFINITE FORM. 207 Also, since wehave andtherefore When thoHO aubHtitutioiiH oremade inthooriginal equation whichgavo {stas],itbecomes -if1-*H.L-i+.s-11 *La-)' ^^^^(.^-i)]1 This iHofthewamo form OHtheequation (A)inthogeneral coso,and inidentical with itwhenwewrite *=- v-1,M=i; andthen therelation between sandxis Now \*=(1 (y)*, ^,9n=(a- )',i/9=(ry-a)";runioinbering that7a$must bepositiveinorder that thu scriesmay convergeovenwhen thovariable isoqualtuunityandassuming thatowgreaterthanj3(whichinpermifwiblo), womaytake Ifitbedesired tohave ^Qpositive, wecanchangethesignofn\ andthen theelements ofthehypergeometricseries are -,^o- 7=1+-.n^2n'n 208 OASES OFSOLUTION [131. andtherelation between sand asis l-sn __(l_a l+sn~ (^ The lattergives ^ 1(1 c) s= (!-)*' andtherefore s=, {1+(1-)*}' while'-*=* (1- )**"= {!+(!- )*} Now thetwoparticularsolutions, when theequationisinits normal form, are <y*andO/-^, andtherelation between thedependentvariable vinthiscaseand thedependentvariable intheordinarydifferentialequationis (116) which becomes y=VX~(z+2~n)(1 cc)~* inthespecialcase. Hence theprimitiveofthedifferential equation doc n is y=Op" (I+(1-aff+C,{1+(1-). Moreover oncomparingthese twoparticularsolutions 5""andaf withthesetofparticular solutions, wefindthatthey correspond toLand in.respectively ;infact,therelations are 131.]INAFINITE FORM. 209 2*11+(!-,))-"......CD and*g-i,-.I-i..}-S-*{l+(l-.)V......(ID 1 thecommon factor os" having boon removed from the latter. Thesu tworolatioim areofcourseequivalent toonoanother. 132.GAME II.From what himbeenproved inthelawtcane itfollowH that,whenweassigntheparticular value 2ton,wohave therelation *_-4^-^ (o^-hl)9 asaHolution of Kmtly.lot *(ft then (I" Suoondly,let |<r, {?,)=[(,,fJ- (f,,fj F-14 210 CASES OFSOLUTION andtherelation is fa=n ff Thirdly, bywriting a=N/3f8, weatoncehave[a;ffl}=3{<r,fj=|/^_^gga\ai a_i where ^= 2^/3' Fourthly,let a-=a?;then Q Now{s,o-}= {5,s8 }= ; ,and . 12o-a sothat Hence k&*V (1*+Sf/T 5*1andtherelation is,=-7=. Fifthly,let f-"gri; then{5,f4}={,W _ (&-!)*' 8'(&-l)''16 274 8(f48-l)" andtherelation is Sixthly,let then 132.] Also andINAFINITE FORM. Fa^""~*211 Hence ',6)= *_ ,,,.,.. fcr.94+2j9as-n8 andthorelation isf.=r=L^-Sw^S-lJ Ittherefore follows thatasolution of inthocasewhun X=1=u.,v=L*(-!) From thisrelation thovuluu of$can hi1foimrl(itisasoinu- whatconiplicatodfunction ofa;}andthencea';andthin will U-iul {02)tothosolution ofthoequation 7 133.CASE III.From thetwoprucodiiigCUSUH anuwonu canboconstructed. For let,inCase II., then byCase I.;andsoff+iy j,i_% l*'^-(!-)' >+a' 142 212 SOLUTION ISAFINITE FORM. [133. Nowchangezinto z>sothat then{s4 IL ,*}= {s,-z]=+_ ____."^(1-*)" 5(1-*) Acomparisonwith thegeneralformula shews that the last relation between zand sisasolution, provided *-> v=$>/*=; andtherefore a== ,7= Hence bymeans oftheprecedingrelation wecanobtain the primitiveof inafinite form. Ex. 1.Shew thatfrom Case n.canbederived inafinite form th& solution of Ex. 2.Shew thatfrom Case in.canbederived inafinite form the solution of Further cases willbefound intheMiscellaneous Examplesattheendof thechapter. Itmay easily beverified that,foralltheexamples given, wehave on taking positive values ofX,p.,vtheinequality thecaseofX+/i+v=lisintegrable bythesimpler method of 68.See Ex. 7,p.126. 134Forfurther information onthesubjectofthehypergeometrio series thefollowing memoirs should beconsulted': GA.TJBS,"Disquisitiones generalesciroaseriem infinitam Ges.Wer&e,i.in.pp.123163; "Detenninatio seriei nostrfflpereequationem difi'erentialeni secundi ordinis," id.pp.207 230. 134]MISCELLANEOUS EXAMPLES. 213 KUMMKR, "Uober diohyporgeonietrischo Roihe," CreUe, txv.pp.39 83aud127172. SCHWA.RZ, "Uobor oinigo Abbildiuignaufgaben," Cretin,t.iiXX.pp.105 120; "Uobcr diojcnigeiiFiille inwolchun dio (Jaiuntltidio hypergoo- motriHuhu Roiho eino'algobraisuhoFunction iliros viortcn Elomontoa darstollt," Crallt),t.i-xxv.pp.202335. CAYLKY, "On tlu>Soliwarzian dorivativo andthoPulyhudral FuuotioiiH," Cumb. 7V7. Traiw. i.xnr.; inthoListofwliich rufcrouuuH willbufonud tofurther memoirs. Tlioro isnlsoamemoir by(IOUIWAT whiohmaybouniiHiiltnd with grunt advantage "Hur I'dquntiimdifTdruntiulk1quiudmut pnur intdgralolaHiSriu hyporguoniiStvicpie" (AnnnlmttieCifnule iiornmlemifitfriaitre,Hdp. rr. t.x.) in which bydevelopingamuthod duunrigimdlytoJauubi hoobtains thoreunite ofKummor andSuhwarz. MISClELLANKOUH EXAJTI'LIW 1.I'rovotliat,if (rtHfr-'-SttficQH^)1rts.-.l11+iJ1UfJM0+2^aO thunArmaybowritten inanyofthoforniH 2.Obtain asolution ofthoequation asahyporgeometricserioM;A,2i,C,D,JE,F&rQ uuppoaedtobeconstants. (Qausa.) 3.Afuuotion issaid tobecontiguoustoF(a,#,y,x)when itisderived from itbychanging oneandonlyoneoftheconstant elements byunity. Let 214 MISCELLANEOUS f(*+l, ft7,*)bedenoted byFa+;F(a-l, fty,*)byFa_ ;and^(a, fty,*> by.F.Then provethefollowingrelations: (i)0=09 (ii)Q=(y-a- (iii)0={y-2a-(j3-a)tf} (iv)0=y{a-(y-0)*}^-ay(l-*) (v)=(y-a-^)JFT+a(l-^)^ a+- (Gauss.) 4.Prove that (1-a?)^(o, fty,a?)Jf(l-a,1-ft1-y,*)-1 -a,i-ft2-yi ). =__ y(l-y) (Gauss.) 5.Bychangingtheindependentvariable inthedifferential equation verify thefollowing equations: (i)(l+y}^F(Za, 2a+l -y, y,y}=ffa, a+J, y, (Gauss.) (ii)(l+y^F(a, a+i^ft jS+i^=^(0,ft2ft (T^-y8). (Gauss.) (iii)^(o, fta+0+isin8 d^F^Za,2fta+j9+i,sin3 |). (Kummer.) Prove alsothat,bychangingthevariable fromxto-&{!+(1-a;)*}"8 , 2a+2 \-^ /aa+12a+2 -4BJ 6' 3' (Kummer.) 6.Shew thatthefunctions PnandQn,which aretheindependentsolu- tions ofLegendre's equation, maybeexpressed byhypergeometrioseries in theforms thevariable xofLegendre's equation being connected with bytherelation . Heine.) EXAMPLES. 215 7.Shew that,iftheindependent variable inLegendre'a equation be restricted tobeleasthan unity,theprimitive mayborepresented by where theaeries,ifinfinite,tirocouvorging. (Heiuo.) 8.DenotingthesoriuB i,a0y,.,a.a+1.0.+1.7.y+1,.,,, 1+^+'"l727O+i:7:H-l*+'"' by^{(a' /fl^ )*}l)rovo thnt /^witinftoH thodifferential equation andobtain twootherimrtiuularHolutioiiH ofthoo<iuAtionintheruwpootivo forms the lirnt ofthusu throo HolutioiiH intunnu ofthoother two(wo 9.Vorifythatauothor nolutiou ofthodifForoutial uquationintholout questionin andhence derive twoother nolutioiui from thoreuultu given inthelout question. 10.Theequation hasaparticularsolution ofthoformx"\determine nandulitaiu thuprimitive. Henceexpresa HIU~IJ;onahyporgoomotricwerioH. (tlourat.) 11.Obtain inafinite form thoprimitive of alsoof (Oournat.) 216 MISCELLANEOUS EXAMPLES. 12.Prove thattherelation as x-\ satisfies theequation Hence obtain inafiniteform theprimitivesoftheequations (i)*(l-.) (ii)*(!-) 13.Prove thattherelation 4a~ 108**(a*-1)* satisfies theequation Hence obtain inafiniteformtheprimitives oftheequations (i)*(i--:)+(*-H) CHAPTER VII SOLUTION BYDEFINITE INTEGRALS. 135.THEprincipal methods which lead toexpressionsforthe dependentvariable interms oftheindependentvariable bymeans ofwhat areordinarilycalled known functions havenowbeengiven; there ishowever another method whichcertainlyleads toasolu- tion ofsome differentialequations thoughthe fullevaluation by theoperationsindicated maynotbecarried out. Thismethod consists inexpressingasadefiniteintegralthevalue ofthede- pendent variable;itschiefapplicationinordinarydifferential equationsarises inthecase ofacertaingeneralclass oflinear equationswhich canotherwise besolved inseries, thoughnotin soconcise aform. Themethod ishowever ofprimary importance inthesolution ofthose linearpartialdifferentialequationsoforder higherthanthe firstwhich arise ininvestigationsinmathematical physics;infact, insomequestionsthese solutions bymeans of definiteintegralsconstitute theonlysolutions hitherto obtained. Here, however, weareconcerned withtheapplicationtoordinary differentialequations. 136.Themethodapplieswithpeculiar advantagetoLinear equationsintothecoefficients ofwhich acenters onlyinthe first degreeandinwhich there isnoterm independentofyorof differential coefficients ofy;suchanequation,initsmostgeneral form,is 218 SOLUTION BTDEFINITE INTEGRALS. [136. where theasand 6'sareconstants. Thismaybewritten d where <f>andtyarerational integral algebraicalfunctions ofthe order ningeneral, thoughtheorder ofeither maydiminish throughthevanishingofsome ofthecoefficients. ToSolve this equation weassume whereTisafunction oftbutnotofas;theform ofthisfunction andthelimits ofintegration (supposed independentofas]areto bedetermined bysubstitutingthisproposedvalue ofyinthe differentialequation.Since theresult ofthesubstitution maybeexpressedintheform fxtP (*)Tdt+$<&^(f)Tdt=0, which mustbeidenticallysatisfied. Theformer oftheterms, being integrated byparts,isreplaced by andtherefore theidentity becomes the firsttermbeingtaken between thelimits oftheintegral,as yetunknown. Now thiswillbesatisfied, ifwemake forallvalues oftincluded within therangeofintegration, and [#*< (t)T]=Q atthe limits. Theformer oftheseequations determines Tasa function oft\thelatter willdetermine thelimits of"thisassumed integral. 137.] DETERMINATION OFLIMITS. 219 137.Toderive thevalue ofTwewrite the firstequation in theform andtherefore whereAisanarbitrary constant. Hence thevalueofyis dt *(*) taken between limitsofintegration defined bytheequation these limits being independent ofas. 138.Wehavenow todetermine the limits. Consider the equation where/^isaconstant. Let^beavalue oftindependentofos andsatisfyingtheequation ;let/*a,...,prbeother constants and /Q'a /9rbecorrespondingvalues oft,allindependentof0. Then ifthevalue * y=- besubstituted intheequation and ifforeach ofthese definite integrals (Tbeing assumed tohave thevalue beforeobtained) asingle integration bypartsbeeffected, asinthepreceding analysis, thenthattheequation maybesatisfied wemust have andwhen this isidenticallysatisfied theforegoingvalue ofyis asolution oftheequation. This lastidentitywillindicate such necessaryrelations asmaysubsist amongthearbitrary constants A, and sowill fixthenumber ofindependent constants-; when this number isthesame astheorder ofthedifferentialequationthe foregoingvalue_ofyistheprimitive, but ifitbeless the necessary number ofparticularsolutions tomakeuptheprimitive 220 DETERMINATION OFLIMITS.[138. must beotherwise determinedExampleswillbegivenhere- after. 139. This isthemostgeneral method ofobtainingthelimits; itincludes asaparticularsetthelimits obtainedbytaking those roots oftheequation which areindependentofas;they obviously make andtheyareusually thesimplest obtainable. When thisequation indicatesonlytwo limits distinct from oneanother, these will givetheonlydefiniteintegral immediately derivable insuchan example. If,however, more thantwo,sayr-r1,limits beindi- cated, then rparticular solutions maybeconstructed; infact, denotingthese limits bya, lt/3a,...,/3r,wederive from them' asthecorresponding partoftheprimitive 9-r(rs \ y=2\A8\e**Tdtl. 8=1(Jo.) Ex. 1.Toapply theforegoing toobtain theprimitive oftheequation __j?_<r},_n d**y-Q- Herewehave -withtheabove notation *(*)--!, Vr(0=11 ;andUierefore -TA6-frdt .*-^o6> or,changing thesignofthearbitrary constant, this is i+i T-A*~\ wMle,maccordance with thegeneral rule, theequation determining the Now this issatisfiedby*=>whenMia"zero andbyt=Qwhen.-A- may^Mthe^^ fthe 139.]EXAMPLES. 221 Itmust benoticed that, justaninthegeneral caaoouoofthedefinite integrals alone wasnotasolution ofthodifferential equation,HOtin's innot asolution oftheequationsince thotunas outaido theintegral aru 3*-: 3Ata-0 iuwtoad ofzero. Thisvalue ofyistliuruforo thoParticular Integral ofthe equation Now thoquantity T(loonnutchange,iffor twowriteat,whuro ina root ofthyequation moreover tholimits ofthedefiniteintegral aruunaltered Hinco inthoequa- tiondetermining those limits thuturin xtinthoexponent haschanged into istatwhioh, HOfarawthinequationIHconcerned, inthoHIUUO anchanging,rinto xu,achange which hasno (ift'uct onthelimitu HJIICO thuyiiruindopundont of .17.Hence wehaveanother dullnitointegralinthuform t-ti /"'j.i+axt /*(*). or,whon thoaimoved outwido thoHignofintegration,itin Forming now those dotinitointagrnlHforalltho(rt+l)"1rooiw ofunity andadding them together wofindanthooxproHHionfory,whioh liantobe bubfltituted, w+l /oo'-+fft ow+1rf+ o When thisvalue isHubntitutod,OHintliogeneral invostigation, thoterms which areunder theintegral sign vaninliidentically and tliat]>art ofthe expression taken between tholiniitn, whioh isfurnished bythointegral involving Ar,iuAr;hence thoreuniting equation, when thinvalue ofyis substituted inthedifferentialequation,is Ifthen thissinglo condition besatisfied among then+l arbitrarycon- stants, thoaboveexpression foryistheprimitive ofthedifferential equation 222 EXAMPLES OFSOLUTION BY [139. Eos, 2.Prove that theabove expressionforyistheprimitiveofthe equation providedtheconstants Asatisfythecondition AQ+AI+A Z+......+An=a Ex. 3.Prove thattheprimitiveoftheequation is,forpositivevalues ofx,given by Obtain thecorresponding primitivefornegativevalues ofx. (PetzvaL) Ex. <LTosolve where aandjareconstants. Here sothat Hence onesolution oftheequation ip taken between thelimits given by Toobtain thelimits, write andsupposeapositive ;thentworoots oftheequationaregiven by t=+q andt=q. Ifnowxberestricted topositive values, athird root isgiven by *=-eo, whilewhenxisnegativeathird root isgiven by t=+ao. 139.]DEFINITE INTEGRALS. 223 Asineither casewehave three values givenbytholimits equation wecan construct twodistinct particular solutions, andsohave theprimitive. Thus whenxispositivetheprimitiveis (t*-q^a~l<Pdt, q while, when soisnegative,theprimitiveis Ex. 5.Verify that,when aliesbetween 2eroand2,theprimitiveofthe equationis o Jo unless abeunity,inwhich casetheprimitive maybewritten y=["efffl)008fl{A+Blog(xsin2 ff)}d6. Jo (Boole.) Ex. 6.Obtain bymeans ofdefiniteintegrals theprimitiveofBessel's equation. 140.Theforegoing generallinear differentialequationisone with variable coefficients which areofthe firstdegreeinthe independent variable; andthedefinite-integralsolution wasob- tained bymeans ofalinear differentialequationofthe firstorder determiningtheunknown function T.Itisnot,however, the onlytypeofdifferentialequationtowhich theassumed form of integralisapplicable ;itis,infact, aparticularcase ofamore general process,indicated bythefollowing proposition. Thesolution, bymeansofdefinite integrals, ofthegenerallinear differential equation ofthentAorder, whosecoefficientsarenotcon- stant butfunctions oftheindependentvariableofdegreenothigher thanm,canbemade todepend uponthesolutionofalineardif- ferential equation oforder nothigher than m,thecoefficients of which arevariable. Thisproposition weproceedtoprove.Let the differential equationbedenoted by 224 GENERAL THEOREM ON whereXr(forallvalues ofthesuffix r)isafunction ofasonly,of degreenothigherthanm,given by r=a, while forsome values ofrsome ofthecoefficients ofthehighest powersofamayvanish. Takingastheparticularsolution the same form asbefore, wewrite with thelimits asyetundetermined, andTanunknown function oft.Now thisvalue ofygives - dor andtherefore theequation,when thisexpressionforyissubsti- tuted init,becomes +f+X^ +......+tZ,+ZJdt=0, which mustbeidenticallysatisfied. Rearrangingtheexpression sothat itmayproceedinpowersofx,andwriting wetransform theaboveequationinto Z7>+ Now theleft-hand side iathesum ofm+1integralsofthe form (pTUjfdti 140.] SOLUTION BYDEFINITE INTEGRALS. 225 andeach ofthese canbeintegrated byparts until thevariable so ceases tooccur exceptintheexponential Thuswehave +(-ir thepartwithout thesignofintegration beingtaken between the limits oftheintegral,asyetundetermined.Denotingtheex- pression byVrforallvalues ofrexceptzero(inwhich casenointegration bypartsisnecessary) andapplyingtheforegoing formula toeach ofthedefiniteintegralsontheleft-hand side oftheequation, we changetheequationinto This willbeidenticallysatisfied iftheunknown function Tbe chosen soastosatisfytheequation =o forallvalues oftbetween thelimits ofintegration. These limits must bedetermined by m~\Vr=0. lJ Now thisequation determining Tislinear with variable co- efficients, and itisoftheordermbut itmaydegeneratetooneof lowerorder; when itissolved, adefinite-integralsolution ofthe original equationisderivable. Hence theproposition follows asenunciated above. Since theequation which determines Tisoforder m,itwill havemindependent particular solutions; thesemaybedenoted by^i^t......Tm.Correspondingtothese there willbem F. 15 226 GENERAL THEOREM ON[140. particularsolutions oftheoriginal equationobtained bysub- stitutingforTin ffTdt thesewvalues inturn. 141. Inthecasewhenm=2theequation which determines Tbecomes or,-what isthesamething, Thefollowingaresome ofthespecialcases inwhich this equationcanbeintegrated very simply. (1)When thecoefficients a,b,caresuchthattheequation issatisfied forallvalues oft;inthiscasethevalue ofTiseasily provedtobe A|= (2)Onmultiplyingtheequation throughout byUt,wecan rewrite itintheform theleft-hand sideofwhich isaperfectdifferential if f77)=TJ(- * i dt*a a \dt dP that is,if at/ dt Ifthevalues ofa,b,cbesuch astomake thisanidentity,then thevalue ofTisgivenby 141.] SOLUTION BYDEFINITE INTEGRALS. 227 which leads totheresult or (3)When theequation inTisreduced toitsnormal formby thesubstitution thenewequationis Asolution oftheequationisatonce obtainable when vanishes, i.e.when. rod/i Further, immediately integrablecases arefurnished when f&is aconstant, orisoftheformX(e+ft)~*,oroftheform\(e Inanycase,whatever betherelations amongtheconstants in thefunctions V,thesolution oftheequation determining Tisof theform while theequation givingthelimits ofthedefiniteintegralis which issatisfiedbythevalues oft,ifany,common to T=0 and^=0._dt Ex.Integrate, bymeans ofadefiniteintegral, theequation wherejuisaconstant. 152 228 SOLUTION BY[142. 142. Another setofequationstowhich themethod ofsolution bydefiniteintegralscanbeappliedisthesetderived from fordifferent values ofn.Tosolve thisweassume where tdenotes anunknown function ofasaloneandPanunknown function ofpalone, both ofwhich functions, aswellasthelimits oftheintegral,have tobedetermined.Differentiatingthevalue ofytwice andsubstitutingintheequation, wefind Choose theunknown function isothat andsupposethatXispositiveandequaltoca ,sothatthedifferential equationis Then theequation which determines tis d* andtherefore -m ifmdenote %n+1.Hence wehave j.atm,ia5m(m 1 1 7^-=and7j-i \ \-tdasx tdoc a? Lettheequation involvingtheintegrals bemultiplied through- outbyaf/mt ;itbecomes, after avery slight reduction, mft*(P*-1)Ptdp- (m-1)fe~*Ppdp=0. 142.] . DEFINITE INTEGRALS. 229 Integratingthe firsttermbyparts,wehave =0. Now thiswillbeidenticallysatisfied ifwemake for allvalues ofpincluded between the limits ofintegration definedby Theformerequationserves todetermine Pasafunction ofp;it isofthefirstorderand linear, and itssolution is m+I P=A(p*-l)~ *, Abeinganarbitraryconstant;andtheequation whichgivesthe limits is The latterequationissatisfied byp=oo,andbyp=Iprovided theexponentofp*1ispositive;thisrequiresthatmshould either bepositiveandgreaterthanunity,orbenegative,and therefore thatnshould not liebetween zeroand 2.Assuming that thiscondition issatisfied, weareinapositiontoconstruct two definiteintegrals;theyare ri m+l e~*(p*-1)*dp, J-i and[e**(p*-Vj~~*^ dp. Theformer ofthese isequalto f\771+1Q711+1 Ie"*"^3 1)*"dp+ Ie~**(^a-1)2m dp,Jo J-i =Ie"1* (p* 1)2mdp+ Ie1* (a 1)Smdp,Jo Jo ri _=(e**+0"")(p 1)2mdp. Jo 230 APPLICATION TOTHE.' [142. Hence theprimitive mayberepresented by fl _*+!(.00 4'(ept+fl-7rt)(pa-l)~2mdp+BI<r"(y-lJo/i substitutingfor twehave /" fflin+l _ +5 /en+z(p'-l)to+4dp, astheprimitiveoftheequation forvalues ofnnotlyingbetween and 2. Ex. Prove thattheprimitive ofthesome equation maybegiveninthe form B >I provided?idoesnot h'ebetween 4and 2. (Lobatto.) ApplicationtotheHypergeometricSeries. 143. Inorder toobtain adefiniteintegralwhich shallsatisfy thedifferentialequationofthehypergeometricseriesweassume y=\(\-vx)mVdv, whereVisanunknown function ofvonlyandmisaconstant; theform ofV,thevalue ofm,andthelimits oftheintegral have tobedetermined. From thisvalue ofyweatoncehave ^=m(m-1)JW(1- 143.] HYPERGEOMETRIO SERIES. 231 sothat,when these values aresubstituted intheequation itbecomes IV(l-vni)m-a[m(m-1)fa(1-*)-mv(1-we){7-(a+ft+1) ffl) Thecoefficient ofA-Vwithin thebrackets isofthesecond degree in 9/i,which isasyetanundetermined constant; letmbeso chosen that this coefficient vanishes, aothat inisgivenby -m(m-1)-in(a+$+1)-a/3=0, or 7?ia+m(+)+a/9=0, whencemmaybotakenequaltoeither aor #.Asthe differentialequationisunaltered when aand/Qareinterchanged, either ofthese rootsmaybutaken;weshall tako m=a, andthen, substitutingthisvalue,wefindthattheequation J7(1-*)*"" [a(a+1)fa+&v{7-a(a-I-/3+vy+1)} t>= must boidenticallysatisfied.Koarrangingthoexpressionwithin thebrackets under thesignofintegrationanddividingoutbythe factor a,wetransform theequationinto JV(1-my*"* (a+1)v(o-1)xdu + IV(1-va)"a-a (vy-/8)(1-IM)dw=0. Integratingthe first terrabypartswehave 'a~1 -Vv(l-v)(\-^)"1" andtherefore theequationbecomes 1 {v(1-v)7}-(ft-.7)Fdv- 232 APPLICATION TOTHE ^ [143. Now thiswillbeidentically satisfied, ifwetake astheequation todetermine F l{vQ.-v)V}-(/a-VY)V andassign,asthelimits oftheproposed integral,values ofvsuch that Tosolve theformerequation, wehave I v Hence v(1-v)F=4/(1-w)Y~ /3 , whereAisanarbitraryconstant;andtheequation [determining thelimits is which, onthesuppositionthat /3ispositive andygreaterthan ,issatisfied byv=and0=1. Ittherefore follows that theequation ofthehypergeometricseries issatisfied by y=AIV-1 (1-vy*-1 (1-<ra)-a do, Jo provided /S6epositive andygreaterthanfi. Itiseasytoshew that,when(1 aru)~a isexpanded andthe coefficients ofdifferentpowersofasareevaluated, theresulting series isaconstantmultipleofthehypergeometric series, this constant factorbeing 144. Ifnowwechangetheindependentvariable fromxto 1sc,thecorrespondingform ofthedifferentialequationis 144.]fHYPERGEOMETRIO SERIES. 233 Asolution ofthisequation (andtherefore oftheoriginal equation) is,fromtheforegoing analysis, givenby provided 13ispositive anda+1greaterthan7.Iftheconditions oflimitation oftheparametersbosatisfied, theprimitiveofthe differential equationofthehypergeometricseries isgiven bythe Humofthese twodifferent solutions. Kr. 1.Obtain intormw ofdefiniteintogralHthocompleteunlutiou oftho equation (HOCKK.2,p.213). Jfo. 2.Prove that, (i)ifbopositive anda+1 greater thany,thenawolutiou IH (ii)ifybogreater than/aand IOHHthana+1, thenaDilution iw y=r*P~l (1-uf"ft-l (l-xuradu; (iii)ifybogreater than/3and a.IOMHthanunity,tlit-n aHolution is i y=fV-1 (1-?t)T-^-1 (1-.vw)-adu. (Ji\cobi.) Air. 3.Obtain thoprimitive ofthoequation 4-g+(*-->-,-<> (whorea/+.a?=l) inthoform IT IT y~Ar(1-^81 andofthoequation inthoform beingtheuamo OHbefore. EXAMPLES. 235 isgiven by y= where theupper signistobetaken ifxbepositive andthelower ifxbe negative. (Petzval.) 3.Prove thattheequation *S-v-odm?' hasasolution given by /"x_jL_ay=BIsm-evrt>dlo') andthatasolution of s y theminus orplus signbeing taken according asxispositiveornegative. Obtain theprimitiveofeachequation. (Petzval.) 4.Investigate theprimitive oftheequation intheform IT_JL^ y=A\cos(oxmsin0)cosm 0c& ./o ir_ l_ ;Icos(opsin0)cos$d<f>,Jo forvalues ofmnotincluded between 1and+1. (Summer, andLobatto.) 5.Shew thataparticular solution of isra ^=^n+i/ (j)2_az )n /a 234.MISCELLANEOUS f.[144. Solve also **--* (iv) 4a;^ (Glaisher.) .4.Prove that,ifre+1bepositive, then *^(1_o-(l-1 isasolution ofLegendre's equation; while,ifnbenegative,asolution is given by 145. Thischapter containsonlyaslight sketch ofthemethod ofsolution ofdifferentialequations bymeans ofdefiniteintegrals ;thereader whowishes forfuller information onthispartofthesubject should consult twoauthorities inparticular. ByfarthemostimportantisPHTZVAL, Integrationderlinearen Differentialgleiakungen; theparts dealing with themethod are 25,of Section n.;1922 ofSection m.; 10,11ofSection v.Theother authorityisEULER,Inst. Colo. Int., vol.ii.,c.s.;thiswork, however, labours under thedisadvantage ofassuming theform ofthesolution firstandthen of findingthedifferential equation satisfied byit.There aretwoothermemoirs whichmightalsowithadvantage beconsulted; onebyLOBATTO, Crello,t.xvii., p.363;andonebyJAOOBI, CreUe,t.Ivi., p.149. Afulldiscussion ofthesolution oflinear differential equations bymeans ofseries andofdefinite integralswillbefound, together withnumerous examples,inaseries ofseparately published memoirs bySPITZER. MISCELLANEOUS -EXAMPLES. 1.Integrate completely theequation 2.Prove thattheprimitive oftheequation 236 MISCELLANEOUS andthataparticular, solution of 6.Shew thattheequation issatisfied by +n /no y=Ifm-l J where!//(#)isgiven by Henoe from thesolution of deduce that of 7.Verify that isaparticular integralofar 8.Shew thatwhen theooefficients ofthedifferentialequation satisfy thecondition^ftj- oa61=6a2 ,thesolution willbe where and thelimits being given by #UiV=Q. (Spitzer.) EXAMPLES. 237 0.Prove thatequationsoftheform mayboreduced tothoform of130,bythoBulmtitutioiiB xm=tandy-^fo; andshew thatA1iadotorminod byaquadratic equation. (Petzval.) 10.Prove thatthoparticular integral of where &denotes x-7-,in y-rrr..... yoyoyu 11.Prove thatthodoiinitointegral flTz**-1 (1-JO""""1"7"x (1-i')*'7"1 (1- ;iiJn ia,when5>/8^0 andf>y>0,aHolutinn ofthedifforentiftl equation Give inthoform ofdefinite iritogmlH thoprimitiveofthisequation. 12.Theprimitiveofthoequation IHy-^f rc-^r+^J(W3 wheren,0,-yarethorootrt of andthearbitraryconatanta arcoonueoted bytheninglorelation D^-JiX'*. (Petzval.) 238 MISCELLANEOUS EXAMPLES. 13.Prove thatthedefinite integral satisfies theequation -3t=m?'fafm~ay. ctea * (Poiason.) 14Prove that Pbeing Legendre'sfunction. (G.H.Stuart.) 15.Shew that,ifftbepositive andalessthan unity, /nP-1(l-n)*-*-"1(!-*) Jo isasolution ofthedifferential equationofthehypergeometricseries. (Jacobi.) CHAPTER VIII. ORDINARY EQUATIONS WITHMOHETHANTWOVARIAB s. 140. IThasalready appearedthat insome cases, thoughthe integrationofseparateterms ofadifferentialequationwould in- troduce newtranscendental functions, thesolution oftheequation OHawhole canbeexpressedinterms ofpurely algebraicalfunc- tions. ThuH, forinstance, theequation _<fe_ +dy=Q (!-)' (I-/)* canbeintegratedinterms ofthetranscendental functions arcwiny\buttheru iHasolution oftheform which isequivalenttotheother.Woarethusnaturallyledto enquirewhether other coses exist inwhich suchanalgebraical relation between thevariables oftheintegralsoffunctions can boobtained when theintegralsthemselves cannot boevaluated without theintroduction ofnew functions. The cose next in pointofsimplicity,which furnishes asimilar example,isthat usually known asEider'sequation,inwhich theobjectisto findtheintegral algebraicalrelation between 01andywhich corre- spondstotheequation where X=a,+bss+cue?+ex6 -\-fx*, and F=a+by+cy*+ey*+f\f. EULER'S EQUATION. Tointegratethisweassume p=x+y, and ^=_*1 dtya;' sothat ^= dtx-y' do7^ jfaandtherefore-f=. dt xy Asecond differentiation withregardtotgives d^Pa_" f^dYdy 1dX Za?)F*X^fdso dv d?~cD-y\2Y*~dy^>i~' 2Z*<fo~^J~ (n-yf \dt~ ~dt thelastfourterms inside thebracketbeingthevalue of o. y~~x' Rearranging andcollecting terms, wehave Ifwemultiply by2andintegrate, weobtain orsubstitutingthevalue for^f 146.]CAUCHY'S METHOD. 241 analgebraicalrelation between asandy,thoughtheseparate integrals requirefortheirexpression ellipticfunctions. Ex. 1.Prove thatanother integraloftheequation Q^X. /r~QTS^"^-* +=o tiL^i-'"V~"*" -STr=T-^r isVJf andverifythetheorem of12inthiscasebyshewing thatthetwoprimitives arenotindependent. ^ L*-OCr^-* IEx. 2.Prove thatanintegral of P*"t*^ 3.Expressinanintegral form therelation between yand ssgivenby 4.Shew thattheprimitiveof maybeexhibited intheform {jf(l-y)(l-J^) where J.isanarbitraryconstant. 147. There isanother method ofproceeding,duetoCauchy; itisquitedifferent froA theformer. Consider ageneral equationbetween thetwovariables ofthe seconddegreeoftheform whereZ,Z15Z,,F,F,Faarealloftheseconddegree,thefirst three inon,andthesecond three iay;thus if F. . 16 242 OAUOHY/S METHOD.[147. p weshould have Then theratio ofdy:dasisgiven by dujdu , 5-dec+5-dy=0. But since u=Y$?+2F^4-F8= ;similarly andtherefore , <fy _Q' adifferentialequation theprimitiveofwhich isu=0. Now since Euler's differentialequationissymmetrical with regardtoxandy,itisnecessarythat itsprimitive u=should besymmetrical withregardto asandyinorder that thepre- ceding analysis mayapplytothepresentcase. Inorder thatu maybesymmetrical, wemust have andX*-XXaisthen thesame function ofasthat7,2-FFa isofy.Inorder toobtain theprimitiveof where X=a+bos+ex?+ess6 +/#*, andFisthesame function ofy,wemust makeXand X*XXathesame. Thecomparisonoftheir coefficients will givefourequationstodetermine the coefficients ofu;but in uthere are fiveindependent constants (there wereoriginally 147.]^CAUCHY'S METHOD. 243 eightasanyonecanbemadeunity, butthreeequations necessary forsymmetryoresatisfied) andtherefore onewillremain undeter- mined andsoarbitrary. TheseAquations givingthecoefficients are =fe'b 4(ft/-al0])- c when thevalues ofthedetermined coefficients aresubstituted in11,theequation u=contains onearbitraryconstant and is thus theprimitive.<J\,,,-tk^1 ,. X-ai*Re. 1.Prove thattheprimitiveof v^'^^^ dxdij___ ^ (9*=yt 0, vrhoro^'a-fla=&A- nftiDV7^. A7 .j,\2.Verify thatthoiirimitivouf (1+a&*-I-"u-' in - where !s-n=-A (Cauulry.) Chap.xiv. i)f(Jayloy'w "EllipticFunctions" luaybooonwultuil with iidvaiitage. 148. IfiiiHtead ofasingle eiiuationbetween two variables, thorelation between which isuxpreasibloinanalgebraical torni, wehaveawystomofn1equationsbetween nvariables, wemay withoutintegrationofeachintegrablo expression rcspruKuiitinan integralform thedependencebetween thonvariables intho shapeofanalgebraical equation;andasthisequationisobtained byanintegrationitmust contain anarbitraryconstant. Tho process made useofinorder toderive itinthogeneraloo,so will buseen todift'ermateriallyfrom thatadoptedinthuparticular case ofn=2. * !()2 244 GENERALISATION OF[148. Letthedifferentialequations be dfljjdx9 dxn_* ~x?"r~x?^+Sz?~>. inwhich forallthesuffixesjj,inthesystem. Let f(x)= (x-ojj(x-a?2)(-aO; 7//\ and let/v (#)denote thevalue ofjwhen init.after thein-ax dicated differentiation hastakenplace, w^issubstituted forx;the value of/'(XP)willtherefore be thevanishingfactoras^x^beingabsent.Solving nowtheabove systemofequationsinorder toobtain thealgebraicalratios of thequantitiesdaol3dxa, ,dxn)wefind Letthecommon value oftheseequal fractions bedenoted by dt,sothatwehave andsoon. The firstofthesegives ~dt)= {/K)}s> andtherefore, after differentiation withrespecttot, 9dx^ d'os^9[~Xl "|dscl3 T. 1.0""~^If * i v\ft I5~~ ~T~_~ It ft*t^-Tn I-.-^H 148.]^EULER'S EQUATION. 245 v9Now Butsince /'W=fa-*,)<X-*,)......(X-O. wehave .,.^-{/' (ajt)l= ,/0*03V7V L-V ,. . 3 I"X]2X 1andtherefore^.-,/v.,= .,\,,- ,3^[_(/ (*i)}'J i/C^!)}11^-^ provided pbenotunity. After substitution anddivision bythe 7JJ coefficient of-j-ontheleft-hand side, theequation becomes 1-19f^i1 ,XX 1XX 1 "* fe,[{/K)}S J+/K)/ (.) ,-^VW/(O^- . 1 Similarly x9rzanz}x*i jra*z.*' i~*3^aLITWl'J/W/W^-*,/W/K)^- and sofortheothers, making?iinall.Now letthenleft-hand sides oftheseequationsbeaddedtogether;thesum willbeequal tothat ofthenright-handsides. Itwillbeseenthatinthelatter, X^X* 1when inther"1expression aterm .\',^-enters, then* - 1 inthes*expression aterm/.//t /./r /x-also enters, and* - thesum ofthetwo istherefore zero. Allthetermscontaining these fractions-will forallvalues ofsandrdisappear;and nr-xt thuswehave 246 GENERALISATION OF [148. We shall afterwards denote0^+3?,,+......+#nbyp,sothat e?ptheleft-hand side is2-73. off 149.Wecanobtain another value fortheexpressiononthe right-handside. LetXdenote thesame function ofxasXiofoslf and letX {/(*)!' beexpandedinpartialfractions. SinceXand{/(0)}aareboth ofthedegree 2?i,there willbeaterm independentofas,which willbeA^\andsowemaywrite Multiplying upby (CDcc^wehave XTf vg ^ ..../=Cj-f5,(a;aj+terms multiplied by(a; a?,)', \J(x)\ ordividingoutbythecommon factors inthenumerator andthe denominator ontheleft-hand sidewehaveG1+B^(oo x^)+terms multiplied by (-*,)>=,_., .. Ifxbeputequaltoas1,theleft-hand sidebecomes(7,andthe right becomes j/1 \ia80 I/(^Jj Theright-handsideoftheequationintheform lastwritten does notinvolve asl}and itspartialdifferential coefficient with regardtoxvistherefore zero;since thetwosides oftheequation areidentically equal,zeromust bethevalue ofthepartialdiffer- 149.] EULER'S EQUATION. 247 ential coefficient oftheleft-hand sidewithregardtox^andsowe have 7\C\ 7\~R 2-1B1+(ao a;^3-1+termsinvolving (so a?j)=0. OflJjOSD1 This istrue forallvalues ofx,andtherefore .^U/MT Similarly Bt=~- withcorresponding expressionsfortheotherquantitiesB.Hence ._1 .u!L^L Lettheequation expressingtheresolution intopartialfractions oftheexpressionconsidered bemultiplied throughout by andlotthecoefficients ofai4""1onthetwosides of beequated.None oftheterms involvingthequantities Ccan furnish terms ofHOhighadegree,since eachbeginswith x*""9 ; each oftheterms involvingthequantities Bbeginswith a?"'1 , andthewhole coefficient from this series ofterms istherefore Since* /()=(as-oO(0-<O......(0-<O- =a"-a;"'1 (tfj+a;a+......+ojj+lower powersofa =a?"poo*"1+lowerpowers, the coefficient ofa?"-1inJ.an(/(a)]"is-2A.J3.That onthe left-hand side isA^;andtherefore ......+B 248 GENERALISATION OFEULER'S EQUATION. [149. Multiplying by~andintegrating,wehave whereEisanarbitraryconstant. But = i4.>+. dt dt dt...... dt __ _ _ "/>.) /tor......f'M} andtherefore theintegral becomes ;.1.Prove thatanintegraloftheequations tfck? ydyzdz ~^+r*+F' where andJPand^aresLmilar functions ofyand ais where (7isanarbitrary oonstant. (Richelot.) Ex. 2.Deduce asecondintegralofthese equations intheform ex)+a (Richelot.) Thetheory ofthese' andkindred equations cannot herebecarried outto thelimits ofitspresent development,asitsoonceases tobelong exclusivelyto differential equations andmergesintothegeneral theoryoftranscendental functions. Thereader whowishes forafuller development onthelines of differential equations than canbegiven here willfindapaper byRIOHHLOT, 149.] TOTAL DIFFERENTIAL EQUATIONS. 249 Crclla,t.xxiii., pp.364r369,veryuseful;aridhewould dowelltoconsult the following papers byJAOOBI, Crdle,t.is.,pp.394403; t.xiii., pp.5578; t.xsiv., pp.2836; t.sxxii., pp.220226, nilofwhich t\recontained inthesecond volume ofhiscollected works. Forthohigher parts, chieflyinconnexion with thetheoryoftranscen- dental functions, thememoirs ofAbel should beconsulted. TotalDifferential Equations. 150.The differential equations withwhich wehave hitherto had todealhave been, exceptin148and149,such asinclude ono.dependentandoneindependentvariable;forthefuture we shall consider those which include more thantwovariables. These maybedivided intotwo classes, oneinwhichonlyonedependent variable occurs, theother inwhichonlyoneindependentvariable occurs. Inequationsoftheformer closaweshallhave thepartial differential coefficients ofthesingle dependentvariablerelatively totheindependentvariables;these arecalledpartialdifferential uquatioiiH and willafterwards bediscussed. Inequationsofthe latter classweshallhave thedifferential coefficients oftheseveral dependentvariables with reference tothesingle independent variable (which maybueitherexpressedorimplied);these are usuallycalled total differentialequations. Now ifwehaveanintegral equation $(>y>z)=c>V where iaaconstant, womaysupposethat #,y,zundergo slight variations dw,dy,dz,which weknow willbeconnected bythe rulation , oyJaz or,ifwoassume thatas,y,zare allfunctions ofsome variablet, then andtheforegoing equationbecomes dtj>dx9<ftdij dcfrdz_'+ ~~' 250 'TOTAL DIFFERENTIAL EQUATIONS. [150. These twoareequivalent forms; theformusually adoptedisthe first;ifinanycasethesecond begiven,itcanatoncebechangedOjO-iOJ intothat ofthe first. Moreover, if^-,^-,^-haveanycommonoxoy02 factor, theequation canbesimplified bytheremoval ofthat common factor;andsowemayconsider thegeneralform ofsuch anequationinthethree variables asrepresented by Pdas+Qdy +Rdz=0, where P,Q,Raregiven functions,ofcc,y,zandareproportional tothedifferential coefficients of$. 151. But, conversely, whenanyequationoftheform Pdas+Qdy+Rdz= isgiven,itdoesnotnecessarilylead toanequationoftheform (x,y,z)=G\ fortheexistence ofsuchanequation impliesthatthethreequanti- tiesP,Q,Rareproportionaltothedifferential coefficients ofsome onefunction, and this isnot satisfied while P,Q,Rarequite general. Wemust therefore findoutunder what circumstances such adifferentialequationwilllead toanintegralofthegiven form;and,ontheassumptionthat suchanintegrali3possible, indicate amethod ofobtainingit. There willremain thefurtherproblemofobtaining asolution oftheequation when theconditionsnecessaryfortheexistence of suchanintegralastheabove arenotsatisfied. 152. Inthe firstplace thenweassume thatsuchanintegral exists;wemust therefore have P,Q,Rrespectively proportional tothepartialdifferential coefficients ofsome function <with regardtoas,y,ztsothatwemaywrite ^=^,pQ^, pR= *$>t ^one^ dydz inwhich pissome function thevalue ofwhich isunknown. From, the firsttwooftheseequations wehave 162.] TOTAL DIFFERENTIAL EQUATIONS. 251 dpJZ dx dp ^-'dy da^dz dp 1f-.doo Multiplyingthe lastthree'equations respectively byR,P,Q andadding, wehave dy] \dx dz which istheequation givingtherelation between P,QandR; and this,whenidentically satisfied, indicates thattheproposed differentialequationleads toanintegraloftheform considered. 153.Weshallnowassume that this relation exists andthat thedifferentialequationtherefore hasaprimitiveoftheform 4>(Bty,s)=Q\ wehtfve toshewhow todeduce thisprimitive. Ifwehadthisprimitiveandproceededtoform thecorrespond- ingdifferentialequationwitharestriction thatzshould notvary, theequationwould be whichequationwould notbeaffected byanyterm intheprimitive which involved zalone. Conversely then, ifweintegrate ontheassumptionthat zdoesnotvary,thearbitraryconstant ofintegrationisaquantity independentofthevariations ofxand yandmaytherefore beanarbitraryfunction ofz.Wereplace thearbitraryconstant byanarbitraryfunction ofzandsohavea relation between x,yand z.Thishowever willnotnecessarilybe theintegral required,foritmaynotsatisfytheequation 252 TOTAL DIFFERENTIAL EQUATIONS.p [153. weonlyknow that itsatisfies theparticularform ofthisinthe casewhen zdoes notvary.Itistherefore desirable toform the differentialequation correspondingtotheintegralintheform in which itnowoccurs;itshouldyieldthegivendifferentialequation andacomparisonofthetwoforms will lead,from thecondition thattheymustbeidentical, toanequationwhich willdetermine thevalue ofthearbitraryfunction ofz.This lastwill alsobea differentialequation;whenintegrateditwillcontain thearbitrary constant inthedetermined function ofzwhich onsubstitution furnishes theprimitive.Hence wehave therule : Lettheequationbe andsupposetherelation satisfied. Integrate asifzwere invariable*, andmake thearbitraryconstantofinte- gration equalto <(z}. Substitute now soastoobtain theori- ginal equation andchoose<$>(z)sothat thecoefficient ofdzisR. Theprimitiveisthenfound. Ex. \.Integrate (ydx+sedy) (a-a)+xydz=Q. HereP=y(az\ Q=x(a z\R=sey }andtheequationofcondition is satisfied. Ontheassumption that zisinvariable thetermxydz disappears andthen a,z-willdivide out,sothattheequation becomes whichintegrated gives xy=A= <t>(z), accordingtotherule. Differentiatingthiswehave j ydx+asdy-jrdz=0. *Ifmore convenient either oftheother variables might beconsidered tem- porarily constant andthecorresponding changes made. 153.] TOTAL DIFFERENTIAL EQUATIONS. 253 luorder thatthetwoequations maybethesamewemusthave d$ xy_ <f) dzaz.az' TT 1fy1 1Hence T-T-=--=- , <f>dz a-z z a, therefore <(z)=C(za), whereCisaconstant;andtheprimitiveis ,vy=C(z-d). * JEx. 2.Verifythat foreach ofthefollowing equations thecondition of integmbilityissatisfied, andobtain theprimitives: (i) (ii) (iii) (iv) (x-a)dx+(z-c)dz+ {A2-(*-a)2- (-c)3 }*dy= ; (v) (vi) (vii) (a:*y-yn- (viii) (2a"+2.iy+Zxf+1)C?A:+dy+2zds= (ix) (2.7?+y2+2.w)dx+Ixydy+a^dz=du. 154.Theprecedingsolution hasbeen obtained onthesup- positionthattheequationofcondition amongthecoefficients of the differential elements das,dy,dzissatisfied;itremains nowto consider the class ofequationsforwhich thecondition isnot satisfied, andforwhich there cannot therefore beasingle general integral. Letusnowassume anyarbitraryrelation betweenas,y,zof theform ^(x>V>z}= 5 thisonbeingdifferentiated gives When theform^isspecified,thesetwoequationswilldetermine zanddzinterms ofon,y,dscanddy(or,generally,oneofthe 254 TOTAL DIFFERENTIAL EQUATIONS.^ [154. variables and itsdifferential interms oftheother twoand their differentials) ;when theyaresubstituted intheequation Pdx+Qdy+Ed&= theymake itoftheform Ndx+Ndy=0, whereMandNarefunctions ofcoandy,thevalues ofwhich will depend upontheform ofthechosen functioni/r.Now thisequa- tionmaybeintegratedandtheintegral, containinganarbitrary constant, willtogetherwith therelation constitute asolution ofthedifferential equation. For itisevident fromthemethod ofderivation oftheintegral that, incombination with-^=0,itfurnishes relations between JE,yandzsuch thatthedifferentialequationissatisfied. Bygivingallpossibleforms totyevery possiblesolution will beobtained. Each solution willbeconstituted bytwoequations. Ex. 1.Solve dz=aydx+bdy. Theequationofcondition isnotsatisfied;some relation between#,y,s must therefore beassumed andthismaybeperfectly arbitrary:letitbe y-/(*). Acombination ofthis-withthedifferentialequation gives dz=af(#)dx+bf(#)dxt theintegral ofwhich ia z=a, If{#)dx+ bf(ss)+C. This, vn.thf(x)=y }forms asolution oftheproposed equation. Ess. 2.Obtain themostgeneral solution oftheequation which isconsistent withtherelation a? Ex. 3.Find theequation which must beassociated with j?a+ya= (/>(z)in order togiveanintegral of {x(x-a)+y(y- 6)}dz=(z- c)(xdx+ydy) ; 154.] TOTAL DIFFERENTIAL EQUATIONS. 255 andthatwhichmust beassociated with >L'^^'-''" ,, v:*.-- lJsoastosatisfy .fifc. 4.Prove that,if/ibeaquantity suchthat ^-,Vc I 3J(*7 T tA-**"^thenasolution ofthegeneral equation mayberepresen 3F _ 9z This isMongers form,^ !r.5.Obtain thegeneral equationswhich constitute thesolution of y<fe-(*-*) (^-ofe). * 155. Itisnotatfirstsightclearhowtheequationofcondition, affects theaboveprocess and, inparticular, whywhat hasbeen' givenasthesolution inthelatter case isnotthesolution inthe former case. Buttherelation between thetwosolutions canbe f"'~*f**~ h seen asfollows.' '{t,ju-U*V Theelimination ofthedifferential element dzbetween thetwo equationsinwhich itoccurs leads totheequation and, inorder that thismaybereduced totheform Mdas+Ndy=0, thevariable z,which occurs init,must bereplaced byitsvalue derived fromty(x,y,z)=0.Now supposetheequationofcon- dition issatisfied sothatP,Q,Rareproportionaltothe differ- ential coefficients withregardtoas,y,zofsome function;ifthis function be^(x,y,z),thenwehave _ dz~QdyPfa.....................} andtheequation involvingdasanddyisidenticallysatisfied. There willthus,onthissupposition,benoother equation necessarilyasso- ciated with theequation ^=0,or,what isequivalentforthis case, T/T=C;thisbyitself issufficient forthesolution ofthedifferential equation,andanyother equationassociated withT/T=Gmaybe 256 TOTAL DIFFERENTIAL EQUATIONS. [155. perfectly arbitrary (suchas%=0),foritsexpressionwillnotenter intothedifferentialequation when formed from theseintegral equations.Ifhowever theequationfirstwritten down benotthat which leads totheparticular properties (A),butbeanother such as^=0,itwill stillbepossibletoderive theequation -fy=C,into theexpressionofwhich theform of^doesnotenter; andwe maytherefore consider asthegeneralsolution ofthedifferential equationtheequation while, ifwewish todetermine yandzseparatelyasfunctions of x,weassociate with thisanyarbitraryrelation betweena,y,z. Ifhowever theequationofcondition between thequantities P,Q,Rbenot satisfied, there isnofunction ^such that the relations (A)hold;andthus Mdx+Ndy= isnotanidentitybutleads toanintegral,theform ofwhich is affected bytheform ofthearbitrary equationfirstwritten down andwhich must beassociated withthatequationinorder tocon- stitute theintegral. Itthusappearsthatthedifference between thetwocases is this;whilewemayconsider that inboth cases twoequationsare necessarytogivethecomplete solution, inthecasewhen the equationofcondition issatisfied oneoftheseintegral equations (called -\IT=(7)iscompletelyunaffected informbytheother(called X=0),butinthecasewhen theequationofcondition isnot satisfied oneoftheseintegral equationsisaffected informbythe other. 156.The difference between theresults inthetwo classes havingbeenindicated, itisnowpossible toadoptamethod of integration which shews thepointofseparation between the processes applyingtothese classes. Let x(a, y,*)= beanyrelation betweenx,yandz;then 156.] TOTAL DIFFERENTIAL EQUATIONS. 257 Wealsohave Pdos+Qdy+Rdz=0. Lettheformerequation bemultiplied byX(aquantitytobe determinedafterwards) andadded tothelatter, sothat or,say, Pfa+Qfoj+Rflz=0. 'Let\besochosen astomakePpQltR^proportional tothe differential coefficients withregardtoas,y,erespectivelyofsome functionty;then theintegralofthelastequationis t(y.*)=G, where isarbitrary, andtheprimitive ofthedifferentialequation isgiven bythotwoequations X(a,y,z}= ) ^(oo, y,z)=G\' Now sincePI}Qt,R^areproportionaltodifferential coefficients withregardtox,ytz,wehave orsubstitutingforPslQltR^andreducing, wehave p\E(BP 3z)+M (dy- __-^U^^ V3* 3/3\3a? 3*/3*V3 IfP,Q,^2bethemselvesproportionaltodifferential coefficients withregardtox,y,z,tho first lineinthisequation vanishes anda solution oftheequationisX= ;Pt,QltR^arethenindependent ofxaiidtherefore $(oo,y,z)isindependentof%. IfP,Q,Rbonotsuch astomake thefirst linevanish, then\ isshownbythisequationtodepend upon thoform of^andthere- fore^alsowilldepend upon theform of^.Theform of-v/rwill inthiscasebedetermined bythemethodgivenin154;butthe foregoing investigationisuseful asameans ofinstitutingthe analytical comparison between themethods. F. 17 258 TOTAL DIFFERENTIAL EQUATIONS. [157. Geometrical Interpretation. 157.Ageometrical interpretationcanbegiventothe differ- entialequationand itsintegral,which will illustrate the differ- encebetween thetwo classes ofequation explainedinthe last twoparagraphs. Ifasusualas,y,zrepresentthecoordinates ofapoint A, theequationwillthenrepresent some locus. LetA'bea pointonthe locusadjacenttoA;thendoc,dy,dzarepro- portionaltothedirection cosines ofAA' andthe differential equation impliesarelation between, these direction cosines;the locus which itrepresentswilltherefore besome curve orfamily ofcurves, andnotasurface orfamilyofsurfaces. 168. Consider nowthetwo differentialequations das' d/dz . P',Q',Rbeingthesame functions ofcc',y',2*that P,Q,Rare ofOD,y,z;theirintegralsareoftheform whereu^andutarefunctions ofas',y',z\andastheycoexist theseintegrals really representtheintersection oftwo surfaces each ofwhich isoneofafamily.This intersection ofanytwo particularsurfaces isacurve, andwetherefore have adoubly infinitesystemofcurves. Onecurve ofthissystem passes through Aand isdetermined bythose values ofa,andaaobtained by substitutinginuvandwathecoordinates ofA.LetA"bethe pointonthiscurve which isconsecutive toA;then thedirection cosines' ofAA"areproportionaltodas',dy,dzortothevalues of P',@,SfatA,that istoP,Q,R.Now thecondition thatAA", AA'maybeperpendicularis which isthegivendifferentialequation; hence itexpresses the factthatAA' isperpendiculartothat curve of(ii)which passes through A.The solution ofthe differential equation 158.] TOTAL DIFFERENTIAL EQUATIONS. 259 must therefore include allthecurves which cutthesystem (ii) orthogonally. Ifwestart fromAinanydirection which isperpendicularto thetangentatAtothatcurve ofthesystem (ii)-whichpasses through A,weshallcome atA'toanadjacentcurve ofthissystem; movingfrom A',.inanydirection atright anglestothisweshall atanother consecutive pointinthispathreach anotheradjacent curve; andsoon.Thepaththus obtained must beincluded in thesolution ofthedifferentialequation ;andasateachpointA wemaymove inanyoneofaninfinite number ofdirections(i.e. inanydirectionlyinginthenormalplaneatAtothecurve ofthe system)itfollows thatthesolution oftheequationwillcontain an arbitraryfunction. Let us,then,drawthrough Aanysurface weplease andlimit ourpathsoastobeinthis surface ;startingfromAatright anglestothecurve of(ii)there will, ingeneral,beonlyone directionpossibleinthesurface andmoving alongthisthrougha small arcweshall atitsextremityAcome toanother curve;at A'there willasbefore beusually onlyonedirectionpossibleinthe surface and itwilllead toanotherpointA'andsoon;andwe shall thusobtain onthearbitrarysurface asingle path passing throughthepointA.HadadifferentpointBonthesame surface (but notlyinginthepaththrough A]been thestarting point there would havebeensimilarlyobtained asingle paththrough B different fromtheformer;andsoforanypoint. Weshouldthereforehaveonanyarbitrary surface asingly infiniteseriesofcurves. 159. This istheexactgeometrical process correspondingto theanalytical process applyingtothecasewhen theequationof condition wasnot satisfied. Forwhatwasthere donewastoassume anarbitraryrelation amongthevariables this istheequationof thearbitrary surface;itwascombined withthedifferential equation and, afterintegration,another equationwasobtainedcontainingan arbitraryconstant which with theoriginal arbitraryrelation was considered thesolution. Thenewequation containingonearbi- traryconstant representsafamilyofsurfaces; andthecombination ofthetwogivesthesystemofcurves which form their intersection. Each ofthese curves liesonthesurface firsttaken, andsowehave 172 260 TOTAL DIFFERENTIAL EQUATIONS.p [159. aninfinite series ofcurves onthis surface. Theprocesstherefore givesthesystemoflineswhich lieonanysurface andwhich satisiythedifferentialequation. 160.Now itmayhappenthatthecomplete systemofcurves (ii)canbecutorthogonally byasurface and sobyafamilyof surfaces; thus ifthesystemwere aseries ofstraightlines all passing throughonepoint theywould becutorthogonally byany spherewhich hadthatpointforcentre. Inthiscaseanycurve drawn uponanorthogonalsurface would cutthesystem (ii)at right angles,since itisatevery point perpendiculartosome oneofthesystem;andsuch acurve would therefore beincluded inthesolution. Hence thegeneralsolution must include all curves thatcanpossiblybedrawn upon anyoneofthese surfaces and therefore,ifwelookuponasurface astheaggregateofall thecurves thatcanbedrawn onit,wemaysaythat thesurface is included inthesystemofcurves. Asthesurface isoneofafamily allthemembers ofwhichpossessthesameproperty, weconsider thattheequationofthisfamilyofsurfaces isthesolution ofthe equation jandwhat hasbeen saidshews ittobethereby implied that theequationsofeverycurve thatcanbedrawnupononeof thefamilyconstitute asolution. 161. Thiscorresponds exactlywith theprocess applicableto thecase forwhich theequationofcondition wassatisfied;wethere had(156)anequation -\Jr=(7andanyotherarbitrary equation %=0,thetworepresentingonecurve oneach ofthesurfaces^r=C; bytakingallpossible arbitrary equations ^=weobtained all possiblecurves onthesurfaces^=0 andthusultimatelythe surfaces themselves intotheexpressionofwhich theform of^did notenter. 162. Itonlyremains toshewhowtheequationofcondition is derivable from thegeometricalconsiderations. Thearguments areapplicableonthesuppositionthatthesystemofcurvesrepre- sented by dx'_dy^_dz' ~P~~~Q~~R' canbecutorthogonally.Ifthey canbecutorthogonally,asat anypoint A,thetangenttotheparticularcurvepassing through <- ".\ -262' i "j^"~)x" ~J f" A* Similarly /9ZA_3Z, v\_x_9y_.3u and XOlCji C|j/ CtoCyOfl/ji andtherefore a-p-\ ^^_^^^ _i_Y (vv^^^] f] Ifthesetofequationsderived from thisbyallpossiblecombina- tions ofthree different suffixes fromamong 1,2,3,.........,nbe satisfied, then the. differentialequationhasanintegralofthe proposedform. The totalnumber oftheseequationsofcondition is%n,(n l)(w 2) ;theyarenot allindependent,forifthere bewritten down thefourequationswhich involve three outofthe* fourquantities X^X^,Xv,Xpanyoneofthem willbefound tobe derivable from theother three. Ex. Prove thatthetotalnumber ofindependent equations ofcondition is i(TO-I) (TZ,-2). 164.When theseequationsofcondition orthenecessarily independent equations areidentically satisfied, theprimitive,which must therefore exist, canbeobtained byanextension ofthemethod adoptedforequations with three variables. Weintegrateasifall buttwoofthevariables were constant andwereplacethearbitrary constant byanarbitraryfunction ofallthose variables which are supposedconstant. Theequationsoobtained isdifferentiated withregardtoallthevariables andtheresult ismade toagree with thegiven equation ;theconditionsnecessaryforthisagree- ment willserve todetermine thearbitraryfunction which was introduced andsotodetermine theprimitive. Ex. 1.Itiseasilyverifiahlethatythecoefficients ofthedifferentials in theequation =0, satisfy theequations ofcondition \vhich arefouriunumber, threebeinginde- 164.] .TOTAL DIFFERENTIAL EQUATIONS. 263 pendent. Followingtheruleweassume thatonlytwo ofthevariables may change andthesemaybetaken tobe#3and#4;theintegralderived is where$isafunction ofo^and#a.Differentiatingthiswehave (-a:B+2#1#4)dxL-x^diOy+x-fdaii=d0, andacomparisonofthiswiththegiven equations shews that "Wethushave anequation involvingthree differentialsdtp,das^dzz, instead offour(weshould have, inthegeneral case,anequation involving n-1differentials instead ofri) ;therule isreappliedtothisandthenumber again dcoroasod byunity andsoon,untilwecanobtain afinalintegral.In theexample speciallyconsidered theintegraliseasilyseen tobe - <jb+A=a*!2+d,VEB2 , whoreAisnowanarbitrary constant;andtheprimitiveis A. Ex. 2.Thefollowing equations haveaprimitiveoftheform considered; obtain itforoach ofthorn : (i)yendx+ziuody+ itaydz+wysdu^Q; (ii) (iii) Equations ofadegree higherthan thefirst. 165. Equations mayarise inwhich thedifferentials ofthe variables occur inadegree higherthan the first; into their solution itisnotproposedtoenterfullybutonlytoindicate a method ofproceedinginsome cases. Thegeneral equationof thesecond degree maybetaken as Xdtf+7dy*+Zdz*+ZX'dydz +ZY'dzdx +ZZ'dxdy=0, inwhich X,Y,Z,X',7',/farefunctions ofx,y,and z.Ifthe left-hand sidecanberesolved intotwofactors, then theequation maybereplaced bytwoothers each oftheform Pdx+Qdy+RAz=0, obtained byequating separatelytozerothetwo factors. The solution ofeither ofthese, obtained byprevious methods, will beaparticularsolution ofthe differentialequation proposed; 162.] .TOTAL DIFFERENTIAL EQUATIONS. 261 Amust coincide with thenormal atAtotheorthogonal surface. Now thedirection cosines ofthetangentatAareproportional to thevalues ofP,Q',R'atA,thatis,toP,Q,Rand if betheorthogonal surface, thedirection cosines ofthenormal at thepoint x,y,z(whichisA)areproportionalto5^,|^,|^sinceoxoyeg thedirection cosines must bethesame forthetwolines,wemust have Pdas Leteach ofthesequantities beequalto/j,sothat d(j> r>90 r\90 r> a*-"^sir'*G3?="*; theelimination of <andfibetween these leads(asin152)tothe equation considered, which istherefore thecondition that the systemofcurves maybecutorthogonally. Caseofnvariables. 163. Inwhat haspreceded only three variables have been supposedtooccur;but itiseasytopasstothecasewhen there aremorethan three. Inorder thattheequation JTjda^+Xa<foa+Z.(2flj, +.........+Xndas.=0, whereX1}_X"a,......... arefunctions ofa^,asa,.........,should have acomplete integraloftheform thequantities X^must beproportionaltothepartialdifferential coefficients ^*- ,sothatwemaywriteJ forallvalues 1,2, .........,nofp.Ifnow \,p,vbethree different suffixes, wehave 264 TOTAL DIFFERENTIAL EQUATIONS. . [161 andthetwogeneral solutions takentogetherwillconstitute th completesolution. Inthecasewhen each ofthelinearequatior issatisfied, inthesense ofthepreceding paragraphs, byasingl integraloftherespective forms ^(x,y,z)-C,=0,^(n,y,z}-C,=Q, thegeneralsolution will, asin19,berepresented by {^(x,y,z)-C}{^(x, y)z)-C}=V.........(A). Inthecasewhen twoseparate equationsareneeded forth solution eachcorresponding pairmustbelooked uponasasolutioi Now thecondition that these should besolutions isthatth left-hand side oftheoriginal equation should beresoluble int factors. Theleft-hand side isequalto andinorder that thismayresolve intotwo factors wemusthave (Y'*-XZ)da*-2 (ZZ'-X'Y' aperfect square,which willbethecase if (Y*-XZ} (X"- YZ]-(ZZ'-X'Y'}*=b, that is,if Z(XYZ+ ZX'Y'Z' -XX*-YY"-ZZIV )= ; or,sinceZisnotzero,wemusthave XYZ+2X'Y'Z'-XX'*- When thiscondition issatisfied thegeneralsolution isobtainec intheforegoingmanner. When thiscondition isnot satisfied theproposed equatioi doesnotadmit ofasingle primitiveoftheform(A)norofase ofseparate primitiveseachgiven byapairofequations ;but i does ingeneraladmit ofasolutionexpressed byasystemo simultaneousequations. Ess. 1.Theequation#W+y%2-zzdsP+2xydxdy= satisfies thecondition;andtheequivalent equations are 165.] .TOTAL DIFFERENTIAL EQUATIONS, which leadtotheintegrals265 andtherefore ageneral solution willbe inwhich, aisanarbitraryconstant. Ex. 2.Solve (i)U'da?+mm'dy*+nridz*+(lm' -fI'm)dzdy+(In1+I'n)dasdz (ii) (iii)dxdydts=Q; (iv) dx, dy, da x}y,ms dxfdy }mdz=0,wheremisaconstant. Ex. 3.Obtain asolution oftheequation a(6- 0}afdydz+b(ca)ydzdx+ o(a-6)zdasdy=0 consistent withtheequation (Theformer isthedifferential equation ofthelines ofcurvature uponthe surface represented bythelatter.) Ex. 4.Also oftheequation afldx, y*dy,s*dz dx, dy,dz x, y,z consistent withtheequation SimultaneousEquationswith constantcoefficients. 166.Wehave hitherto consideredonly singledifferential equations ;weproceed nowtotreat ofsystemsofequations. The simplest andatthesame timemostfrequently occurringclass is that inwhich there isitonlyoneindependentvariable ofwhich all other variables which occur arefunctions; fortheseparate andcom- pletedetermination ofeach ofthesedependent variables, thenumber 266 SIMULTANEOUS DIFFERENTIAL EQUATIONS. [1 ofequationsinthesystem must beequaltothenumber ofdep dent variables. Inthis class areincluded most ofthedifferen equations ofdynamics ;thus inthecase ofthechiefproblem physical astronomythatofdeterminingthemotion ofasysten material bodies under theinfluence oftheirmutual attraction there isasingle independent variable, thetimeelapsedft some definiteepoch,while thedependent variables arethe ordinates oftheseveral bodies; these coordinatesvary with timeand sofurnish thevarying positionsofthebodies, and t' areindividuallydeterminate since thenumber ofequatiom equaltothetotalnumber ofcoordinates. Allequations deal with thesmall oscillations inamoving systemofbodies are i included; inthem there istheadditionalsimplification that equations arealllinear, thequantities multiplying thedifferen coefficientsbeingconstants. Thegeneral theoryofthelatter willbefirstconsidered. 167. Let tdenote theindependent variable andDstand d/dt ;takingthesimplest possible general case,weshallhave equations involvingtwodependentvariables denotedbyBOan Astheequationsaresupposed linear, alltheterms invoh differential coefficients of a;canbegathered together, andso. forallthoseinvolvingdifferential coefficients ofy;andtheeq tionsmaytherefore bewritten intheform ^ where/j,/,, fa, aarerationalalgebraical integral functions \ constant coefficients and2\and 3?aareexplicit functions oftal aconstant orazerovalue notbeing excluded.Operate onI thesides ofthefirstequation with <B(D)andonboth theside thesecond with(f>1(D) ;thentheybecome .CD)TJ T' Since thefunctions haveonlyconstants intheir coeffici itfollows that 167.] SIMULTANEOUS DIFFERENTIAL EQUATIONS. 267 andtherefore theaboveequations give {0S(D)/ (.D)-fc(D)/ s()}=&(D)2;-0,()!TB......(II). Now let lita,ml}raabetheindices ofthehighestdifferential coefficientsin/^/j,, <j>lt<arespectively;then theindex ofthe highestdifferential in <8(D)/ (D)isma+Z,andin&(D)f, (D)is wZj+la;ofthese twonumbers letndenote thatwhich isnotless than theother, sothat nistheorder ofthehighestdifferential coefficient of a;intheforegoinglinearequation determiningx. Tosolve itweadoptthemethod ofChapterui.applicabletoan ordinary single equation ;ifPbeanyvalue of aswhich satisfies theequation (therecalled theParticularIntegral), and\,\,...,X,, thenroots oftheequation *,(*)/, 00-*i (>)/,00=..................(A), thecomplete value ofxis whereA 1,At,............,Anarearbitraryconstants. Proceed inthesamewaytoeliminate onfrom thetwofunda- mental equations byoperatingonthe firstwith/a(.D)andsub- tractingitfrom thesecond after thishasbeenoperated uponwith /,(D) ;wethenhave {&()/ (D)-ft(D)f t(!>))y=/(D)Ta-f t(D)T,......(Ill), andsoasbefore whereBl}Ba....... ,Bnarearbitrary constants, andQistheParti- cularIntegralofthedifferentialequation (III). 168.Wehave intheexpressionsforthetwodependent variables twosetsofconstantsarisingfromthedifferentialequations II.andIII.;theyarebothcomposedofarbitrary constants, but wedonotknow whethertheyareindependentofoneanother; thisdependence mayexistandyettheconstants maybearbitrary. Thus anyoneoftheconstants Bmightbeamultipleofoneof theconstants A;the latterbeing arbitrarytheformer would besoalso.Wetherefore must determine thenumber ofinde- pendent arbitraryconstants. Todothis letthevalues ofxandy besubstituted ineither oftheequations (I),sayinthe first;then 268 SIMULTANEOUS DIFFERENTIAL EQUATIONS. [168. thetermsinvolving PandQwhich areparticular integrals give ontheleft-hand sideatermTtwhich willcancel with thatonthe right-handsideandtheresulting equationis 001J*+(AJ t00 Since this istobesatisfied forallvalues oft,wemust have the coefficient ofeachexponential zero,andtherefore sothateach constant Bcanbederived from each constant A. Thenumber ofindependent arbitraryconstants inthecomplete solution ofthesimultaneousequationsistherefore n,Le.theexpo- nent ofthehighestindex intheoperator ^(DJ-fcW)/.^). Hence thesolution oftheequations (I)isgivenbytheforegoing values ofonandy\thequantitiesXoccurringintheexpressionsare theroots oftheequation (A),andtherelations between fhecon- stants aregivenbyequations (B). 169. Inexactly thesamewayitmaybeproved that, ifthere bethreedependentvariablesgivenbythethreeequations thenumber ofindependent arbitraryconstantsenteringintothe completesolution istheindex ofthehighest powerofDinthe determinant /8CD), 170. Iftheroots oftheequation (A)whichgivethecoefficients oftintheexponents berealandunequal,thesolution givenabove iscomplete.Itremains toconsider thecases 170.]SIMULTANEOUS DIFFERENTIAL EQUATIONS. 269 (i)when there isapairofimaginaryroots; (ii)when there isapairofequalrealroots; thecase ofequal imaginaryroots willfollow fromacombination of these two. Fortheformer thesolution obtained remainsgeneral, but itis desirable tochangeitsothattheformmaybefreefromimaginary quantities.Thetwoimaginary roots, say\and\,maybedenoted byay9i ;henco thecorresponding parbofa;is e^(A/tt+Ate-^) i that is,eat (Ltcos/3t+Zusin/3t), onchangingthearbitraryconstants asin44;thepartofycorre- spondingtothetwoimaginaryroots issimilarly eatMcos Instead ofmakingthenecessary changesinthe relations between AandB,itisbetter tosubstituteagaintheseexpressions inoneorother ofthefundamentalequations andderive thecorre- spondingrelations asbefore. Forthelatter casethesolution obtained coases tobegeneral because twoconstants, sayAlandAa,become mergedintoone; but itmaybeproved, exactlyasin 44,that thepart of as depending uponthisrepeatedroot\is andthepartofyis Ex. 1.Prove that inthelatter casetherelations between thefourcon- stouts reducingthorn totwoindependentconstants are Ex. 2.Ifanimaginaryroota+fti"berepeated,writedown thecorre- sponding partsofthecomplementaryfunctions inxandy. 171. Itmayhappenthat thequestioninconnection with which thedifferential equationsarise will afford some indication oftheform ofthe result. Thus inaproblem relatingtosmall 270 SIMULTANEOUS DIFFERENTIAL EQUATIONS. [171. oscillations weshouldexpectthevalues ofthedependentvariables tobeexpressedinterms ofpurely periodic functions; and itwould thenbepropertosubstitute forxandyrespectivelyfunctions oftheform ijcosft+Lasinft, M1cosft+H^sinft, instead ofe**intheequations (II)and(HI). Byequatingtozero thecoefficients ofcosftandofsinftineachequationafter these values have been substituted there willbefourequationslinear andhomogeneousinthequantities LandM;andtheeliminante ofthese willfurnish thevalues of$.Ifontheother hand the problemindicate amotion ofunstable character theform ofvalue for asadoptedwould be * (L^cosft+Lasinft), -and sofory\but ifthere benoexternal information ofthis character thentheordinary method should beadopted. Ex. 1.Solve theequations dx_ dy Herewehave andtherefore theequationforxis aothat x=Acos<at+BBin.a>t. Similarly y=A'coBut+JB' smut. The relations between A,B,A',B1areatoncederived bysubstitutingir thefirstequation:wehave -o>Asin<ot+toBoos<at=-a>A'cos a>t-mB1sinat,t or A'=-B, andB'=A. Theshortest method would havebeen tousethe firstequation'togive? interms ofx,sothat =_ldxy~ o>dt =Asinu>tBcos cat. Thismethod ishoweverapplicable onlyinparticularcases. 171.] SIMULTANEOUS DIFFERENTIAL EQUATIONS. 271 P Ex. 2.Solve theequations dPx dy -\---a-+atf=0 When wecollect theterms which belongtotheseparate variables, the equations are ;::) Hence theequationforxis andthevalue ofxis x=L1cosfat+Lzsinfat+Iiscosfat+Z4sinfi$ , wherefa3andfaaaretheroots oftheequation (u*-p)*-a*p=0; andthevalue ofyis y=M:sinfat-Mzcosfat+M3sinfat-M^cosfat. Itiseasytoprove thattherelation between theconstants is Ex. 3.Solve dm They mightbesolved byadoptingtheordinaryrule;thefollowingis another methodapplicabletothisform. Multiplythesecond equation bymandaddtothefirst;then -r(x+my)=x(a+ma')+y(b+mb')+o+mo' provided mbesochosen that b+mb'=m thatis,ifmbearootoftheequation (a-b')m-b=Q. 272 SI1OJLTANEOTJ8 DIFFERENTIAL EQUATIONS. [171. Theforegoingdifferential equation being itsintegralis Letmandm^betheroots ofthequadratic equation; then this isan integral provided miseither m^orm^.Onsubstituting m=m1'W6have andonsubstituting m=mtwehave (a+mpT) (us+in&)+o where J.xandAzarearbitraryconstants. These twoequations constitute the completesolution ofthegiven pairofsimultaneousequations. Ex.4Solve inthesamewayasthelastexampletheequations Eat. 5.Solve thefollowing equations: (i) (ii) (iii)4g+9+44^+49/=i, 3^+7J (vi) (vii) -3^- SimultaneousEquationswith variablecoefficients. 172. Itwillbeassumed asbefore that there isonlyone independent variable andthat therefore thecoexistence ofm simultaneousequationswill suffice todetermine therelations be- tween themdependentvariables andthat ofwhich each isa function. 172.] SIMULTANEOUS DIFFERENTIAL EQUATIONS. 273 Further itwillbesufficient toconsidersystemsofsimultaneous equations which areonlyofthe first order, fortothese anyother system canbereduced. Thus ifintoanyoneofagiven system adifferential coefficient ofthe 71thorder should enter, such as j~,wocould obtain anequivalentseries ofequationsofthe first orderbymakingthesubstitutions dy dy dy.. - ' = which are alloftheorderstated; andthecorrespondingsub- stitutions for alldifferential coefficients oforderhigherthan unitywill transformanysystemofsimultaneousequationsof anyorder intoanequivalent systemofequationsofthe first order. Ifthere bemdependent variables, wemust have inthissystemmequationseach oftheform Z/1'2/B> >ymtda' lfaj=' 173.Thesolution ofthissystemofequationscanbemade to depend uponthesolution ofasingledifferentialequationofthe 771thorderconnectingone ofthodependentvariables with the independentvariable. For letthemequations besolved soastogivethemdif- ferential coefficients asexplicitfunctions ofthe variables, and supposethese relations tobe dy. 'da" . (".ft. ft........ 2/J> Letthe firstofthesebedifferentiated m1times insuccession withregardtoa?,andafter each differentiation andbefore thenext letthevalues of-^,,-jf*besubstituted from the laat ClCC wD F. 18 274 SIMULTANEOUS DIFFERENTIAL EQUATIONS. [173. m1oftheseequations.There willthusbeobtained, including the firstequation, mequations connecting dx' dx*''dxm ' with thevariables x,y,,y3, ,ym;from thesemequationslet them lvariables y^ya, ,ymbeeliminated, andthere will result asingle equationwhichmayberepresented by ') Thisequation beingofthem^order has(8)mindependent firstintegralseachinvolvingonearbitrary constant,allthem constants being mutually independent; andtheseintegralswe mayrepresent bytheequations inwhich theconstants Careindependent. Butfrom thepre- ceding equations weknow thevalues ofthedifferential coefficients ofy,interms ofallthevariables;when these are'substituted in thesetofequations F,thelatter taketheform which aresufficient todetermine each ofthevariables yasa function ofa;;theyareanintegral system andcontainmarbitrary constanta Hence wehave asthegeneralresult : Thecompletesolution ofasystem ofmdifferential equations oj thefirstorder bet/weenm+1variables dependsonthatofaw 173.] SIMULTANEOUS DIFFERENTIAL EQUATIONS. 275 ordinary differential equation ofthemthorderand consists ofin equations, connectingthem+Ivariables and containing min- dependent arbitraryconstants. 174.Theforegoingisthegeneral theory;"butinparticular casessimplificationsariseenabling much ofthelabour indicated inthegeneral theorytobedispensedwith. Thus, iftheequations consist ofaseteach ofwhich islinear,itmayhappenthatan integralofeachequationoftheform canbeobtained intheform andthelong processwould notneed tobegone through. Again, instead ofdeterminingthemindependentfirstintegralsit would besufficient todetermine theprimitiveoftheordinary equationofthe ?/ituorder, forfrom itcould bederived other m 1equationsinwhich thevalues ofthedifferential coefficients could besubstituted, andanequivalentresult would besoderived. Again,inthucasewhon theequationsare alllinearwecansolve them toobtain theratios ofthem+1differentials intheform X which mightbucalled thesymmetricalform;themode oftreat- ment forthese willHomutitnea(depending \ipontheform ofthe denominators inthose fractions)differvery materially from,and bomuch more convenient than, thegeneral process. Examples illustrative ofthiswillbefound appended. Ex. 1.Thogonorol method canboavoided,ifintegralsofallbutcue equationconboobtained aiid,dfortiori,ifallthointegrals canbeobtained. Tims theequations Idjs+indfi+iids=0, xdx-\-ydy+zdz=Q, leadat01100 totheintegrals which determine yandzinterms ofx. 182 276 SIMULTANEOUS DIFFERENTIAL EQUATKJNS. [174 Ex. 2.Solve.Vtr*^" -t)dt. Jy2dx dt tdt' __ _. ,dx da/dz ,Ex. 3.Solve T=T= -^= -^,-where Inequationsofthisform itisconvenient tointroduce somenewinde pendent variable andmake allthosevariables, whichalreadyoccur inthe equations given,functions ofthisnewvariable.Callingthelatter twemaj assume, asanadvantageous form, dtdx_dydz 7=T~F=Z __Idx+mdy +ndz~ provided Z,m,n,Xbesochosen that al+a'm+a"n=\l bl+b'm+b"n=\m cl+o'm+<f'n=\n thevalue ofris Id+md'+nd". Kliminating I,m,nbetween these three equations, wehave aX, 6,a, b'-\, c'.a" b" c"-\=0, acubic equation determining X;letitsroots beX1}Xa,X8.When Xxissub stituted inanytwooftheforegoing equations theratios ofI :m :ncanb derived;letthem bedenoted byZL:m^:^andsuppose thecorrespondin{ value ofrtoberx;with similarexpressionsfortheother values ofX.Thei forthevalue\wehave dt_ Zjflfo+m^dy+n^dz theintegral ofwhich is 174.] SIMULTANEOUS DIFFERENTIAL EQUATIONS. 277 - Similarlyc2 and<^t Inorder toobtain thegeneral solution ofthesystem ofequations asgiven wemust eliminate tbetween theseequations; whenwewrite o1=AG.i=J5o3 whereAandBarearbitrary constants, thegeneral integralasrequiredis givenbytheequations .i ,-i l=A Ex. 4.Solve inthismanner theequations , dy dz! Ex. 6.Thismethod may alsobeappliedtosolve certain systems of equationsinwhich thevariables donotoccur sosimplyasinEns. 3.Thus letusconsider whore T,T^Tawofunctions oft.Multiplyingthesecond equation byIand addingittothefirst,wohave providedIandXaredetermined tosatisfy theequations a+/a'=X, HOthatthevalues ofXareX}andX.j,thetworoots of (a-XX&'-X)-a'&=0. Theintegraloftheforegoing oquatipn being (x+WeWU^A+tt^+lTJ^Mdt, thecomplete solution oftheoriginal equationsisgivenby Ex. 6.Solve thesystems ofequations ..dx .2,w+ <(-i!- l^<278 II ..,*SYSTEM OFEQUATIONS ,'(0)It-j-=m7i(3j-ef Xt"*^^ ir CW 9lc""^s^771{SO~VJ '' . diz?[174. =Iz fix Aspecial system ofequations mDynamics. 175. There aretwo classes ofsimultaneousequationswhich areextremely important;one isthe classalreadyconsidered in 148,149 asthegeneralisationofEuler'sequations leadingto thehighertranscendental functionsordinarilycalled Abelian functions;theother isthesystemofequations which determine themotion ofaparticleattracted toacentre offorcewhich acts accordingtothegravitationallaw.The lattermayberepresented bythesimultaneousequations -(i),= = _ df~ do;'d?~ dy'W~ Tz inwhichRisarationalalgebraicalfunction ofror(a;9+y9 thedistance ofthepoint x}y,zfrom theorigin. Toexpressthe complete integralthreeindependent equations (ortheirequivalent) willbenecessary. Since eachequation maybereplaced bytwo oftheform ck ;d'dcol ~dt3E fa' givinginallsixequationstodetermine thesixquantities,the investigationof173shews thatwemust have sixarbitrary constants inthesolution. 1"75.] INDYNAMICS. 279 Ifwemultiplytheequations (i)by-5-,-J-,-^respectively, add andintegrate, wehave d* inwhichBisanarbitraryconstant. Another formmaybegiventotheequations (i).SinceRisa function ofrwehave dR=dRdr asdR decdrdoc~rdr' andsofortheothers;andthus(i)becomes dfa asdRdfyydR d?z zdR Therefore ofwhich twoonlyareindependent;theintegralsofthese are respectively dy dm -. *7t-yTt=c<' dz dy~"TSi-" dao dz~ Z-j2~CD-j7=G.. dt dt8 Squaring andadding thesewehave dec 280 SYSTEM OFEQUATIONS[175. whereAisanarbitraryconstant;this isequivalent to that is,to dt andtherefore rdr /=/ From theequation justobtained wehave *++(*), andtherefore ~d/r'dA==~ l*di+did?' that is When thisvalue issubstituted inthemodified form ofthe original equations,the firstofthem is d dV a, or ordfda; dr\ ..ac .r-a!+A=0 Let dd>=ATr Adr thentheforegoing equationfor-is .* i(^+?<), 75.] ^INDYNAMICS. 281 ndtherefore a/-=alcos <jt+aasin$.....................(iii). Thesecond andthirdequations similarlytreated lead to in^.....................(iv), z-=Glcos <j>+c,sin <.....................(v) ; ndinthese theconstants a,b,carearbitrary. Buttheyarenot idependent ;forwehavealways * */ whatever bethevalue of<,andtherefore (a,"+b*+c^)cos"+2(a^+bj)a+c^,)cos sin<f> +(aaa+&aa+c^Bin1=1=cos9+sin" 3satisfied forallvalues of0,sothat (vi). Thesixconstants areequivalenttothreeindependentconstants. further, wemayput (iii)intotheform = />tcos(<+ ), vhereptand^arearbitrary constants, andthere isthus associated vith anarbitraryconstant andonewillnotrequiretobeadded ntheequation f Adr.,..> =-.(vil). Jrter'(R +B}-A*}* 'Wehavenow sufficientequationstodetermine thegeneral ntegral. Bymeans of(vii)isgivenasafunction ofr,and .herefore by (ii)asafunction oft;hence(iii), (iv), (v)giveootytz isfunctions of t.Moreover wehave sixindependent arbitrary :onstants, viz.,A*,B,aandthesixquantitiesaltaa,&,,&a,cltc, :onnected bythethree relations(vi). These therefore constitute .hegeneral integralofthedifferential equations. 282 MISCELLANEOUS EXjiMPLES. Ex. Solve inthisway ' +/*-<>) i Also solvebytransformingtopolarcoordinates. MISCELLANEOUS EXAMPLES. 1.Prove that,if d6(inncos<=d$ (m-n cos0) /IN / 0+d> 1 then 2m-nfca+^J+n fcooa-^--cos--H =0, abeing anarbitraryconstant. 2.LetF(x) denote theintegral [ dx Jo{(!-) (!-**)}*' provethatthealgebraicalrelation equivalentto 3.LetE(#)denote theintegral verify that E( where altxi}#3arerelated asintheprevious example. 4.Verifythat =0 isanintegralof dx, dx dx, being givenbytherelationafl+ys=1. Interpret theresultgeometrically. (Oayley.) MISCELLANEOUS EXAMPLES. 283 5.Prove thattheintegralof i (l+2/3)dx^^-dlL _ maybeexhibited iutheform (l+a>) whereaisanarbitrary constant;andthatof mayboexhibited intheform whero/and J"aredefinite constants andaiwanarbitrary constant. Shew thatthegeneral integralof where -i'=(^ ^w, iifec, I)3 , 3'=(Ji Z,.TO,7% I)3 , is XYZ~{k+l (x+y+z)+m whero #=( I,m, and ziaanarbitrary constant. (AlacMahon and 0.Prove thatintegral relations equivalentto eldd<t>d+=-,^ sin8 where AX{(1-Kain2x)(1-Xsin11x)(1- /*iJ^x))*> aingsin^+r-r (sin* 6-sin"0)(sina6-urn*^)" (8ma </>-siua6}(&na-ai sin sin5COBi/rAi/r. "~ ' ' , cos-fycos<J3sindAd COB6cos^sinrRU 'J""sin20)(sinTfl-sin"^j+ (sina-sin26}(aw?i-si Determine AandBfrom theconditions that<=aand^=j3when5=0. 284 MISCELLANEOUS EXAMPLES. 7.Findtheprimitivesoftheequations (i)(ay-bB)da;+(cz a3;')dy+(bx cy')ds=Q; ...dx(y+e-2x) dy(g+a-9y) de(x+y-2z) w(y-*)(a-:0^(-y)(*-y)+(*-)(y-) (iii) 8.Obtain theprimitiveoftheequationc.vtJr. -7*.*"' (aP-yt+s^dx+zdssty-x^&dy --(y2-^3 ) ^\8ZjG~fV\-.ll^t*- intheform* '-UNC,.<a'' 1 ^- ^*. r> where x=ue. Sn('VT' -^^'-uo ,.\ * f r 11 9.Solve thesimultaneous equationsr'l" Iif' >;, expressingeach ofthequantities x,y,saseUiptiofunctions. 10.Integratethesystemofequations -=Q,dt (jq+ oo>--j byBUint+bs cosni=0, 6woosw+o^;sinnt--at=0,at inftf-bxoosnt+ay-j; 11.Integratethesimultaneous equations =0 - where fiswritten foroos(at+6)and77forsin(at+o).3Tu* tftC"* (LiouviUe.) MISCELLANEOUS EXAMPLES. 285 12.Solve thesimultaneous equations 13.Shew thatanysystemoflines described onthesurface ofthesphere #a+ya+8|a=7'2andsatisfying theequation (l+Zin) vdx+y (1-#)dy+zdz=Q would beprojected ontheplaneofxyintoparabolas. Find theequation oftheprojectionsofthesame systemofcurves onthe planeofyt. 14.Shew thatMonge's method (Ex. 4,154)would,ifweintegratefirst withrespecttoonand ^ipresent thesolution oftheequationinthepreceding exampleintheform11 (0 Applythistosolve theproblemofthepreceding example andidentifythe results. 15.Integratethesimultaneous equations whereRisafunction of(x1*+xf+...+#n2 )*' (Binot.) rl.f\wwv^ Kti-i.r, CiCi t.: .^i. , iVV> * j';- ' ..>"r ..*&. *tKr,,.. '4H^lk I U J/C^A / CHAPTER IX. PARTIAL DIFFERENTIAL EQUATIONS OFTHEFmsT ORDER. 176.HITHERTO wehavebeenconsideringforthemoatpart differential equationinwhich thedependentvariableor,inthe case ofaaetofsimultaneousequations,variables aresupposed bo befunctions ofonlyasingle independent variable; wenowproceed, toconsiderequationsinwhich thenumber ofindependent variables isgreaterthanunity, and shallsupposethatthere isonlyasingle dependentvariable. The latter isusually denoted byz;ifitbea function ofonlytwovariables these areusually denotedbyasand y;ifzbeafunction ofmore than two,sayofn,then itiscon- venient todenote the latter byxisa?a,#8,o?n.The first; dz partialdifferential coefficients intheformer case, viz.,^-and OSS de j-,arerepresented bypandqrespectively ;inthelatter casethe a partial differential coefficients =,~, ,=arerepresented f dfljjd#aoccn respectively bypltpy,pn. Anequationinpartialdifferential coefficients isarelation, between theindependent variables, thedependentvariable (whicli isanunknown function ofthose variables) and itspartialdifferen- tial coefficients withregardtothem;itisofthe firstorder- when thepartialdifferential coefficients which occur arealloforder nothigherthanunity,ofthesecond orderwhen thepartial differential coefficients ofhighestorder which occur areoforder two;andsoon.Inthischapter weshall consideronlyequations ofthefirst order. 176.] PARTIAL DIFFERENTIAL EQUATIONS. 287 Itmayhappenthatwehavemore than asingledifferential equation relatingtothesame setofvariables; forinstance we mighthavetwoequations between z,as,y,p,q.Inthiscasethe twoequationscould besolved andfromthem values ofpandqin terms ofsc,yand2could bededuced;these could besubstituted intheequation dzpdx+qdy, andweshould thus obtain atotal differential equation. Similarly inthecase ofnindependentvariables nequationswould besuffi- cientandnecessarytodetermine pltpt, jpw;these nequations would thenbeconsidered asfurnishingatotal differential equation. When thenumber ofequationsislessthanthenumber ofpartial differential coefficients and therefore ofcourse lessthan the number ofindependent variables, wearenotable todeduce from them atotal. differentialequation; usually wehaveonlyasingle equation givenandwothen call itapartial differential equation. AHinthecaseofordinarydifferentialequations,theintegration oftheequationiuthederivation ofallthevalues ofzwhich when substituted inthedifferentialequationrender itanidentity. Classification ofIntegrals. 177. Beforeindicating methods ofintegrationandgivingsuch classes ofequationsasareeasily intugrable,itisnecessaryto classifythedifferent kinds ofintegralsofapartialdifferential equationandtoprovethattheclauses include allpossible integrals oftheequation. Forperfect generalitythepropositionsshould beprovedforanequation involvingnvariables, butthuproofsaro givunforanequation involving onlythroe variables;this limita- tionhastheadvantageofshorteningtheequations andoflessening their number, while theslightestconsideration willshew that it ispossibletopasstothegeneralcase without anyessential difficulties ofanalysis. 178.Supposethatwohavebetween z,SDI}#a,XBarelation of theform /(*,a,vo. fl,aa)=(1), 288 CLASSIFICATION OFINTEGRALS OFA [178- inwhich ax,aa,aaarearbitrary constants andwhich contains nodif- ferential coefficients ofz.Toobtain pl,pt,pgwehavetheequations Sf Sf+=.(2). Betweenequations (1)and(2)thethreearbitraryconstants canbeeliminated; ifin(1)there weremorethan three arbitrary constants theseequations would notbesufficient fortheelimina- tion, while ifthere were fewer than three there would bemore than sufficientequations. Lettheresult oftheelimination in thepresentcasebedenoted by F(PVPVPV*,*VB**J=...............(A), which willbethepartial differentialequation correspondingtothe integralrelation(1). Conversely,thisintegralrelation(1)isasolution of(A),andit contains threearbitrary constants. Wecannotexpect more than. threearbitraryconstants inasolution of(A) ;for,onpassingfrom. suchasolution tothedifferentialequation bythemethod inwhich. (A)hasbeen obtained from(1),onlythree constants couldbe eliminated. Hence(1)contains thegreatest number ofarbitrary constants thatwecanexpectinasolution of(A). ThenameComplete Integralofanequationisgiventoa relation between thevariables which includes asmany arbitrary constants asthere areindependent variables. 179. Thesupposition hasbeenmade thatavavasarecon- stants andwehavededucedequation (A)from(1)and(2).But wemaysupposethatavavatarefunctions oftheindependent variables; iftheybesuch astoleave unaltered theforms of Pi>PvPvthen *nedifferentialequation obtained bytheelimination ofthese functions willbethesame asinthecasewhen thequan- tities awerearbitrary constants, formerealgebraical elimination willtakenocognisanceofthevalue ofthequantity eliminated butonlyofitsform.Now with thenewsuppositionthatthe 179.] PARTIAL DIFFERENTIAL EQUATION. 289 quantitiesaarefunctions ofthevariables oovcovopa,thevalues of thepartialdifferential coefficients aregivenbytheequations 3* 3a;a 3a Buttheforms ofpltpa,paaretobethesame asbefore when they-weregiven byequations (2) ;inorder that thismaybethe casewemusthave _ a a3a3aa3o>83aa.(3). LetRdenote thevalue ofthedeterminant 3ct, 3ffln 3tt_ sothattheforegoing equationsareequivalentto "53TM^>"ol~=v>-tt2' .(4). Now ifEdonotvanish these canonlybesatisfied by and these arethreeequations which determine thevalues of c&j,aa,a flinterms ofthevariables. The relation(1)isstilla solution withthechangeinthequantities a;when thevalues F. 19 290 CLASSIFICATION OFINTEGRALS OPA [179- r justfound aresubstituted forthemwehave asolution of(A) which contains noarbitraryconstant. This solution moreover will obviouslydiffer fromasolutioncontainingnoarbitraryconstant butderived from(1)byassigning particularconstant values to fflj,aa,Bin(1) ;thustheresult ofeliminatingthearbitrarycon- stants between(1)and(B)givesanew solution. This solution iscalled aSingular Integral;itisarelation between thevariablesinvolvingnoarbitrary constant, but itis notaparticularcaseoftheComplete Integral. 180. Theequations (4)will allbesatisfied ifR= ;andas wearenowassumingthata1}avaaarenotarbitraryconstants but functions ofthevariables, thisequationwillbesatisfied bya functional relation between al}aa,aa;this functional relation ,j*>maybearbitrary,sothatwemaywrite ''a^fa.aj (C), inwhich <>denotes anarbitraryfunction.Multiplying nowthe equations (3)bydocltdas^,dx6respectively andadding, weobtain Butfromequation (0)wehave O_L O_L_ (/u) 7 O(D 8 9a,19aa /2J 2^" "3J.\ /OjC sothatdf ,dfty\j ,fW ,9/9< ^-+^-^ }da.+ (^-+^-- oal9a83a/]\9aadaada Sincec^andaaareindependent,their variationsda,andda3art alsoindependent;inorder that thisequation maybesatisfied w< must therefore have 3 da 3a fl Theseequations (C)aresufficient todetermine at,ava8ii terms ofthevariables andtheexpressionssoobtained willinvolvi thearbitraryfunction<f>;whentheyaresubstituted in(\),th.. solution takes anewformwhich isdifferent fromboth oftheothe two. 180.] PARTIAL DIFFERENTIAL EQUATION. 291 This solution iscalled theGeneral Integral;itisarelation between thevariablesinvolvingtwo(or,inthecaseofnvariables, Ti1)independentfunctions ofthose variablestogetherwithan arbitraryfunction ofthose two(orn1)functions. Theequation R=could alsobesatisfied bymaking agan arbitraryfunction ofalone orofa 1alone, sothatweshould thus arrive atdifferent classes ofGeneralIntegrals ;butthese areall lessgeneralthan theformer, inwhichonlyasingle arbitrary relation between allthequantities aoccurs. This iseasily seen from theconsideration thatif,inequation (C),aabeexpanded in powersofa 1thecoefficients arearbitraryfunctions ofaa,while ifT/T(tfj),anarbitraryfunction ofaltbeexpandedinpowers of a^thecoefficients aremerely arbitraryconstants;andthelatter is obviouslyincluded intheformer. 181. Itisthus manifest thatwehave threefundamentally distinct classes ofsolutions ofpartialdifferentialequations; it remains toshew that there arenoothers, and this willbedone byprovingthefollowingtheorem : Everysolution ofthedifferential equationisindiided inoneor other ofthethree classes ofsolutions oftJieequation which are constituted bytlieComplete Integral,theSmgular Integral, andthe General Integral. Let(A)representthedifferentialequation, and(1)theCom- plete Integralofthisequation;then theequations (B)and(0) willgivetheSingularandGeneralIntegrals ;letanyother solu- tionoftheequationberepresented by ty(z,osltaoa}- (4). As itisconvenient toapeakofeasexplicitly expressed in terms oftheindependent variables, weshalluseZtorepresent the value ofthedependentvariable derived from'(l) and torepresent thevalue derived from(4).This lastequation gives 9i/r ,9^-f) -\i f\yJt8Sfl'1 192 292 CLASSIFICATION OFINTEQBALS OFA[181.r Ifnow,wemake these values ofthedifferential coefficients agreewith thosegiven byequations (2),wehave thethree equations _ dzdXj_dz dz(5); andthese determine thevalues ofa,,cia,aainterms ofxv,#B andthedependent variable. Now since(4)isasolution ofthe differentialequation, we have F(Pi>P*>PB>&nv a)" ; andsince(1)iaasolution, wehave satisfied, when thequantities aarearbitrary. The lastequation isalso satisfied when thequantities a,instead ofbeing arbitrary constants, become functions ofthevariables, provided these functions aresuch astoleave theforms ofpl}p^,paunaltered; andwemay therefore replace thembythefunctions ofas^oia,a;8obtained as their values from theequations (5),providedthenecessarycon- ditions besatisfied. When this isthecasethevalues ofpltpa,pB arethesame forthetwoforms oftheequation (A);andwethen havefromacomparisonofthese twoforms thenecessary equation where inZtheconstants al}aa,a8arereplaced bythevalues that havebeen derived forthem. Inorder thattheforms ofpltpisptforthenewvalues ofthe quantitiesashould beunchanged,thethreeequations ofthe form dzdtO dzdzl d(V J/^.9/30,^ dzV9^^ da:,daad^Tdaa 181.] JPARTIAL DIFFERENTIAL EQUATION. 293 must besatisfied atthesame time as(5);andtherefore thevalues ofai;a9,asaresuch astosatisfytheequations . 3aa Butthose areoftheform oftheequations (3)which enable usto passfrom theComplete Integraltotheother twoIntegrals;hence thevalues ofaareincluded amongthose whichgiveeither the Complete,theSingular,ortheGeneralIntegraloftheequation. And anthenecessaryconditions havebeen satisfied, wehave -5 orthevalue ofzderived from thegivensolution coincides with thevalue derived fromoneorother ofthethreeprincipal integrals. Thisprovesthetheorem andshews that thethree classes adoptedinclude allpossiblesolutions. Ifonsolvingtheequations (5)thequantitiesa,befound to boallconstant, then thegivensolution willbeaparticularcase of thoComplete Integral ;iftheybefound tobefunctions ofthe variables andthoro exist afunctional relation between them of theform ct8= <j>(aB>aj, them thegivensolution willbeaparticularcase oftheGeneral Integral;iftheybefound tobefunctions ofthevariables and there bonosuch functional relation between them, then thegiven solution istheSingular Integral Ex. 1.Assuming thattheComplete Integralofz=>pyis \a investigatethenature ofthesolution 4s-2xy=(^+y2 )seoa+ (jfl-#B )tan a. Ex. 2.Assuming thattheComplete Integralof&=px+qyis logz=a log#+ (1-a)logy+6, 294GEOMETRICAL t[181. investigate thenature ofthesolution Ex. 3.Assuming thattheComplete Integral ofepz+qy+pqis investigatethenature ofthesolution z+xy=Q. 182. Inthecasewhen there aretwoindependentvariables- andonedependent, thethreemaybetaken asthecoordinates of apointinspace ;andtherelations between theseparate integral* canbeinterpreted geometrically. TheComplete Integral, beingarelation betweena,yand z,is- theequationofasurface andthisequation includes twoarbitrary parameters;sothattheComplete Integral belongstoadoubly infinitesystemofsurfaces, ortoasinglyinfinite system offamilies- ofsurfaces. Thisintegralisoftheform <(a,y,s,a,&)=0. Inorder toobtain theGeneralIntegral wemake oneofthe parameters anarbitraryfunction oftheother, say6=6(a),and eliminate abetween $(a>,y,z,a,6)=' b=0(a) Thisoperationisreally equivalenttoselectingfromthesystem offamilies ofsurfaces arepresentative family andfindingitsenve- lope.Ifaparticular familybetaken (which occurs when 6ismade adefinite function ofainstead ofanarbitrary function), thenthe equationofitsenvelopeisaparticularcaseoftheGeneralIntegral. Theforegoing equationsastheystandrepresentacurvedrawnon thesurface ofthefamily whose parameterisa,while theequation resultingfrom theelimination ofabetween them isthenvelope ofthefamily ;hence theenvelopetouches thesurface represented bythefirsttwoequations alongthecurverepresented bythethree equations.Thiscurve iscalled thecharacteristic oftheenvelope ; 182.]INTERPRETATION. 295 andtheGeneralIntegralthusrepresentstheenvelopeofafamily ofsurfaces, considered ascomposedofitscharacteristics. Inorder toobtain theSingular Integral,weeliminate the parameters between theequations <(a,y,z,a,6)= fi.dda 9#= dbU Thisoperationisthesame asfindingtheenvelopeofallthe surfaces included intheComplete Integral;thethreeforegoing equations givethepointofcontact oftheparticularsurface represented bythe first ofthem with thegeneral envelope.The Singular Integralthusrepresentsthegeneral envelopeofallthe surfaces included intheComplete Integral. Butwhen theelimination hastakenplacesoastoleave a relation betweenas,y,and z,itisnecessarytoensure that the resulting equationisthat oftheenvelopeandnotthat ofanyof the lociwhich areincluded inthesameequations.Such loci are, forinstance, thelocus ofconicalpointsand the"locus of double lines, neither ofwhich satisfies thedifferentialequation. Itistherefore desirable tosubstitute theresult (whenitcannot at onceborecognisedastheequationofanenvelope)inthediffer- entialequation ;itistoberetainedonlywhen itisasolution. Itmayhappenthat theentire systemofsurfaces does not admit ofthisgeneral envelope ;insuchacasetheSingular Integral willnotexist forthecorrespondingdifferential equation,and its non-existence willbeindicated bytheequations ordinarilyused to obtain it.Examplesofthis willhereafter occur. Asanexampletoillustrate theprecedingdiscussion ofthegeometrical relations between theintegrals,consider theequation ass+.by+oz= (a?+&2+ca )i=l........................... (i), which contains twoindependentconstants. Itiseasytoprovethat the correspondingdifferential equationis andthatthegeneral envelopeofalltheplanescontained in(i)isthesphere l................................. (ii). 296 DERIVATION OFTHESINGULAR INTEGRAL[" Hence(ii)istheSingular Integralof(A),andthesphere represente (ii)touches each oftheplanes represented by(i)inapoint. Toobtain theGeneralIntegral weeliminate abetween +/(>/<-> inwhich /(a)isanarbitraryfunction. This isclearly theenvelope familyofplanes theequation ofwhich containsonlyoneparameter ;ai istherefore adevelopablesurface. Theequationofanydevelopablesur which envelopesthesphere,isthus included intheabove General Inte Theprocessofmaking6afunction ofaisequivalenttodrawing on spheresome definite curve;andthedevelopablesurface istheenvelope o tangent planestothesphereatpoints which lieonthis line. 183.Theexplanationof 179shews how theSing Integral maybederived from theComplete Integral ;itis,1 ever, possibletoderive itdirectlyfrom thedifferentialequa asisthecaseinordinarydifferentialequations. Forthesake ofbrevity, supposethat there areonly independentvariables. Lettheequationbe ^fay>z>P>s)= > ofwhich theComplete Integralis F(a,y,z,a,b)=0, where aand barearbitrary constants; theSingular Integr obtained bycombiningtheequation F=with =0....................... (Ada dbv SinceF=0 istheintegralofthedifferentialequation thev ofz,p,qderived from theintegralwillrenderi/r=aniden andthesubstitution ofthevalues ofpandq(butnotthat derived fromF=Q willingeneralrenderT/T=equivalenttc integral equation.Let this latter substitution bemade, so pandqarereplaced byfunctions ofa,y,z,a,b;then inordi findtheSingular Integral wemust form theequationsanalo to(A),whichequationsare d^jrdp d-^rdq_ dpdadqda~' (fydp _ dpdbdqdb~' 183.] FROM THEDIFFERENTIAL EQUATION. 297 Theseequations maybesatisfied intwoways:firstly, bywriting ^T=n-?i. dp dq' secondly,ifand donotvanish, then da9636da The latterequation implies arelation oftheform which doesnotinvolve either aor6,butmayinvolvequantities!' multiplying aand 6intheexpressionsforpandq;that is, quantities dependingonat,y,and z.Ifboth thearbitrarycon- stants occur inpandq(whichdoes notalways happen) the equation $=wouldimplythattheyareeffectively only one, or thatoneofthem isafunction oftheother; theequationsused thengivetheGeneralIntegral,with which wearenotnow concerned. Wethusreturn to theelimination ofpandqbetween theseand-|r=willfurnish a relation betweenao,y}z,which isindependentofanyarbitrary constant. Ifthisrelationsatisfythedifferential equation,itisthe Singula/r Integral ;andwhen therelation isfound bythismethod itisnecessarytoseewhether thedifferentialequationissatisfied. Thereason that thisprecautionisnecessaryissimilar tothat which renders thecorresponding precaution necessaryinthecase ofordinarydifferentialequations ;when thesurfaces represented haveanenvelope,thisenvelopewillbegivenbytheequations Butthesesameequationswillbesatisfied bythecoordinates of anypinch-point ononeofthesurfacesrepresented bythecomplete integral ;thelocus ofthesepinch-points, however, iseasilyseen nottobeasolution oftheequation. Theequationswillalsobe satisfied bythecoordinates ofanypointPatwhich twodifferent 298 THESINO-ULAB INTEGRAL.f [183. surfaces ofthesystem touch, andtherefore bytheequationofthe surface "which isthelocus ofthesepoints. But thissurface has notnecessarilyforitstangent planeatPthattangent planewhich iscommon tothetwosurfaces, andtherefore thevalues ofpandq (which givethedirection-cosines ofthetangent plane)derived from thisnew locus arenotthevalues ofpandqwhich satisfy thegiven equation ty=0.Such alocuscorrespondstowhatwas before called thetac-locus(28) ;and,while itmaynotbetheonly locus(otherthan theenvelope) which isintroduced, thepossibility ofitspresencerenders necessary anenquirywhether theequation between as,y,2satisfies thedifferentialequation. Ess. 1.The differential equation hasforitscomplete integral (a?-acosa)8+(yasina)B+8s=XaaB , Xbeing supposedadeterminate constant. Formingtheenvelopeofthissphere bytaking F=(x-aoosa)a+(y-asina)2+e3-X2az=0, weeasilyfind ittobe Nowtaking andfollowingtherule forderivingtheSingular Integral from thedifferential equation, wehave ?X=2p#-2X2s(x+pe)=0, The lasttwoequationsaresatisfied byz=Q,which thoughfreefrompandq isnotasolution ofthedifferential equation.Infaotbydrawing afigureitis easilyseenthate=0 isatac-loous, being theplanewhich contains thepoints ofcontact ofthedifferent non-consecutive sphereswithoneanother obtained bygivingallpossible values toaand a. Ex. 2.Consider thesystemofcones 300 LA-GBANGE'S [184- differentiate withrespecttoeach oftheindependent variables andhave du du\ dd>(dvdv\--- -= j du\dx*dz) dv\dx 3rf>fdu du\ 3d)fdvdv\_ du\dy" dz/ dv\dy andtherefore fdu dv which, onrearrangement, gives where'du du\fdv ,dv (ii), E or,what aretheequivalentsofthese, du .(iii). *5rty5r-"5~OSDayoz Hence, whenwehave adifferentialequationoftheform(ii), intowhich the differential coefficients enterlinearlywhilethe quantities multiplyingthesemaybeanyfunctions ofas,y,z,wehave acorresponding integral given by(i),provided wecanobtain uand. vinorder toinsertthem inthatintegral equation. Adifferential equationofthisform issaid tobelinear; thedifficultyinthe solution isthederivation ofthefunctions uand v. 185.Now letusconsider theequationsu=aandv=6,where aand 6arearbitrary constants, and letusform thedifferential equations correspondingtothem.Wehave du ,du ,du -,_ft dot"bydz dv , .dv , .dv 801 or -#- (-) These arethe differentialequationswhich have fortheir integralsu=aand v=&;theycanbeformed atoncefromthe coefficients inthedifferentialequation. Wethushave thefollow- ingrule*: Toobtain anintegral ofthelinearequation writedown thesubsidiary equations dco_dy_dzF~Q~' andobtain twoindependent integrals ofthelatter;letthese be u=aandv=b. Thenanintegral ofthepartial differential equationisgiven by <f>(u,v)=0, where<j>denotes anarbitrary function. Anarbitraryfunctional relation between uandvofanyform. willbesatisfactory ;thuswemighthave u=-^(v), wherei/risanarbitraryfunction. 186. This rule enables ustoobtain anintegral involving an arbitrary function;itwillnowbeshewn that itisthemostgeneral integral possible,inthat itincludes allsolutions ofthedifferential equation.Let ^(a),y, )= *Thetheoryoflinear partialdifferential equations-was firstgiven byLagrange, aswell astheclassification oftheintegralsofequationsofthe first order. The subsidiary equations (iv)aresometimes called Lagrange'a equations. 183.] ,LAGBANGE'S LINEAB EQUATION. 299 inwhich m,6arearbitraryconstants;thecorrespondingdifferential equation iseasilyobtained. Theequations, which givetheenvelope,are sin6(xacos0)-cos6(y-asin0)=0, m These areallsatisfied by ,m whichgive but zisarbitrary. Theequationsarealsosatisfied by 2az=, andthecorrespondingeliminant is The lastequation representstheenvelope ;thedoublyinfinite systemof oones isgenerated bytherevolution, round thedirectrix ofaparabola,ofall therightcircular oones whose vertices lieonthetangentatthevertex tothe parabola, andoneslant sideofanyoneofwhich coincides withthetangentto theparabola drawn throughthevertex ofthecone. Theequation isthat ofthecylinderonwhich lieallthe(singular)circles which, aretheloci ofthevertices ofthecones intherevolution round thedirectrix. Forfuller information onthesubjectoftheSingular Integralsofpartial differential equationsofthe firstorder amemoir byDABBOUZ, M&noires de I'Insttiut deFrcenoe,t.xxvii. (1880),should beconsulted. Lagrange'sLinearEquation. 184.Wehave seenthatamongtheintegralsofadifferential equationthere isone theGeneral Integralintotheexpression ofwhich anarbitraryfunction enters; thededuction ofthe differential equationfrom theintegral impliestheelimination of thisarbitraryfunction. Thesimplestformpossibleforanintegral ofthisnature, when there aretwoindependent variables, isthe equation <j>(u,v)=Q........................ (i), inwhich isanarbitraryfunctional symbolanduand vare definite functions of an,yand z.Inorder toeliminate $we 302. LAGRAtfGE'S [186. r beasolution oftheequation Pp+Qq=E, andletthesolution ofthisequationobtained bytheforegoingrule be<p(u,v)= ;thenfrom equations (iii)wehave Since-x/r(a>,y,z)=0,wehave thesubstitution ofthese values ofpandqinthedifferential equations gives PQ+Q?+IiS=0. das ayoz "Wehave thus threeequationslinear inP,QandR\when thesequantitiesareeliminated wehave 3-^r d^r9^=0. 9^' 9y'aF du du du dao'dy'de dv dv dv dx'dy'de Hence there issome definite functional relation betweenty,u,v; letitbe ^=F(u,v), whereFisadefinite function. The solutionty(x,y,z)= is therefore thesame as and, sinceFisadefinite while<f>isanarbitrary function,, this solution isincluded in *(,iO-o, that is,isincluded inthesolution obtained bythemethodgiven intherule. 186.]LINEAR EQUATION.. 303 * This latter solution isthus themostgenwalsolutionpossibleof thisform;itevidently correspondstotheGeneralIntegral 187.Corollary.Theequations14a=and vb=are integrals ofthedifferential equation. Forthegeneralsolution may bewritten u=^(IF), wherei/risanarbitraryfunction. Take then^r(v')=av,where aisanarbitraryconstant;theequationthenbecomes ua=0, which isthe firstofthestatedintegrals. Similarlyforthesecond. These results canbeobtainedindependently. Theforegoing article shews that, inorder that$(on,y,z)=maybeanintegral wemusthave dec Buttheequations areactuallysatisfied;hence ua=andvb=areintegrals. 188.Wethus seethat,when there isasingle arbitraryfunction entering simply (that is,without anyderivatives) intoanintegral equation,thecorrespondingdifferentialequationisnecessarily linear; andthatthelinear differentialequationhasforitsmostgeneral integralarelation intowhich anarbitraryfunction enters. We therefore infer that, inthecase ofadifferential equation which is notlinear, thearbitraryfunction which isessential totheGeneral Primitive cannot enter inamanner similar tothat inwhich the arbitraryfunction enters intheforegoing equation ;infact,with it willbeassociated intheGeneral Primitive itsfirst differential coefficient. 189. Intheforegoing wehave limited ourselves tothecase oftwoindependentvariables;theproofofthemethod when there arenindependentvariables follows theformer onexactlythe same lines,andthecorrespondingrule is : 304 EXAMPLES OFLAGRAJJGE'S[189. Toobtain themostgeneral integral ofthelinearequation writedown thesubsidiary equations ^j_^a_ _^n_dz prp~.~........."p~n~R} andobtain nindependent integrals ofthese;letthem be Mi=ai.ws=aa........... un=an. Connect thesequantitiesubyanarbitrary functionalrelation thisequationistheintegral required. Theproofofthis, aswellasthatofthecorresponding corollaries, viz. thatul=a l,ua=av.........,un=anareintegrals ofthe equation,isnot difficult. Ex. 1.Solve theequation xp+yq=z. Lagrange's subsidiary equationsare dx_dy_dz as~ y~ e' ofwhich, twointegralsarez=ay, e=bx; henoe thesolution oftheequationis Itcanbeexhibited intheforms 3 ./e\ ,e- ]and- y y which three areeasily seen tobeequivalenttooneanother. \ \ Eso. 2.Solve theequation ixis)q= l/ymx. Lagrange's subsidiary equations are dx_dy_dz mxny~ ptx-lx~lymas' Hejioe aada;+ydy+ede=Q, whence and Idx+mdy+ndzQ, whence lic+my+nz=b ; andtheintegraloftheequationis Ix+my+ra=(a?2+ya+s?). 189.] LINEAR EQUATION. 305 1 Ex. 3.Solve theequations (i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (is) Jfo?. 4.Solve theequation (fsa+xa+e)p 1+(xs+ Lagrange's subsidiary equations ore Each ofthese equalfractions dz-dx1_dz= -(-ai)~ ~-(e- Theintegralsofthese are 2"i37i S^rf/n Z~~ i2>o andtherefore theintegraloftheequationis where&stands for +x+xz+aj3. Ex. 5.Prove that inthelastquestion, if,whenz=Q}thevariables be connected bytherelation thentheintegralis {(^-a)s+(a?a-z)B+(^ B-)B }4(a'i+^+^a+2)3=(^i+^+^8-32)3 . (Mansion.) Ex. 6.Solve theequations (i) (ii) a ^(iii)a?2a;3zpl+a?3a?1p2+37^3^)3= F. 20 306 STANDARD[190. Standard Forms. 190. Beforeproceedingtoindicate amethod ofintegration which isapplicabletothemostgeneral equationofthe firstorder, itisadvisable tonotice afewstandard forms ofdifferential equationswhich admit ofintegration byveryshortprocesses and tooneorother ofwhich many equationscanbereduced;asthe generalmethod isusually muchlonger than thatwhich iseffective foranyofthese standard forms, itisadvantageoustoseewhether theequationisincluded under oneofthem. 191.STANDAED I :Equationsinwhich thevariables donot explicitlyoccur;suchequations maybewritten intheform f(P,2)=0, Asolution ofthis isevidently z=cue+by+c, provided aand6aresuch astosatisfy Ifthen thevalue ofbderived from thisequation be6=/(a), theComplete Integraloftheequationis TheGeneralIntegral andtheSingular Integral must inthe caseofevery equationbeindicated aswellastheComplete Integral, ortheequationisnotconsidered tobefullysolved. Equationswhich donotexplicitly come under thisstandard canoftenbeincludedbychangesofthevariables;thus forinstance functions ofwwhich occur intheequation might admit ofassoci- ation with thepandfunctions ofywiththeq.Butthechanges needed foranyequationcanbedeterminedonlyfortheparticular circumstances oftheequation ;there isnogeneral rule, since an equationcannotalwaysbereduced tothisform. Ex. 1.Solvepq=k. Theforegoing shews that s=aas-srby+o 191.]* FOEMS. 307 isasolution provided db=k theComplete Integral therefore is kz=ax+-y +c.a TheGeneral Integralisobtained byeliminating abetween theequations where isarbitrary. TheSingular Integral,ifitexist,isdetermined bytheequations -y+o(L 0=x-^y 0= 1 thelastequation shews thattheSingular Integral doesnotexist. Ex. 2.Solve pq=afnyn . Thiscanbeputintotheform Let dZ=s-*l de,sothat(1-$Z)Z= dri=yndy, andtheequation becomes which isincluded under thelastexample. Em. 3.Solve theequations: (i) (ii)a lm {iv)fP*sec8x+qncosec2"y=zm~n ; (v) (vi) j31 (vii) 202 308 STANDARD .[192. 192.The differentialequationsincluded under theform haveanimportant interpretation when viewedgeometrically. We know thattheequationofthetangent planetothesurface atthepoint ff,y,is andthesurface istheenvelopeofthetangent planes. Now if o JJiET between -^and%-there bearelation df Of] 9W\. . or orr allthequantities 17,s,75-arefunctions ofasingle quantity, Aof 077 andtherefore there isonlyasingle parameterintheequationof thetangent plane. Theenvelopeofaplanewhose equationis ofthisform isadevelopable surface, andhence thesurface con- sidered isadevelopablesurface. Ittherefore follows that isthegeneraldifferentialequationofafamilyofdevelopable surfaces; andtheequivalentGeneral Integralistheintegral equationofthefamily. 193.STANDARD IL Inattemptingtoreduce anequationtotheprecedingstandard wemayfind itpossibletoremove from theequationtheindepen- dent variables, sothattheynolongeroccurexplicitly;but it maynotbepossibletoremove thedependentvariablelandthe equationwillthenbeoftheform Weassume asatentative solution *=/(; 193.]FORMS. 309 (beingwritten instead ofCD+ay),inwhich aisanarbitrarycon- stant.Wethenhave dzddz dzd& de=^7*.~=a~ andthesubstitution ofthese intheequation gives dz dz This isnolongerapartialdifferentialequation,asthere isnow onlyoneindependentvariable. Thisindependentvariable does notexplicitly occur, andthustheequationconies under Standard IV.(18)ofordinarydifferentialequationsofthe first order. dz Solvingfor-^wehaveanequationoftheform dz,. .=9(z>a)> thesolution ofwhich is or x+ay4-b=F(z,a). This istheComplete Integral ;theGeneral andtheSingular Integrals maybefound bytheordinarymethod. Ex. 1.Solve theequation 9(^8+2*)=4. Ifwemate thesubstitutions BISinthestandard case,theequation becomes or theintegralofwhich is theComplete Integraloftheequationtherefore is 310 STAtfDABD-l [193. TheGeneralIntegralisobtained bytheelimination ofabetween (z+a*)*={3i+ay+6(aW \andSa(e+az^={iD+ay+d(a)}{y+ff(a)}i where 6isanarbitrary function. Itisnotdifficult toprovethatthere isnoSingular Integral Ex. 2.Solve theequations: (i)*W(l-); (ii) 2*y*=z(a-px}', (iii)p(l+q*)=q(z-a); (iv)1=Pzpa+p6p1i (v) 194. The relation between theintegral andthe differential equation admits ofageometrical interpretation. The firststepin theprocess ofsolution iswritingfor as+ay,which isequivalent toturning theaxes intheplaneofocythroughanangle equalto tan"1 a,andmagnifyingthecoordinates inthatplaneintheratio of(1+a8 )*:1.Itisthenassumed that zisafunction of but isindependentofthecoordinateparalleltothenew axis ofy.Now representsacylinder whose axis isparalleltothenew axis ofy; andtherefore theequation givesthecylinders satisfyingthiscon- dition. Butnow, returningtoouroriginal axes, since aisan arbitrary constant, thea. offisanarbitrarylineintheplane, andtherefore also isthelinetaken forthetransformed axis ofy. Itthus follows thatwhatwefindbyourprocessofintegrationwill beallthecylindricalsurfaces with axes intheplaneofasywhich satisfythegivendifferentialequation. 195.STANDARD Ed. Inattemptingtoreduce agiven equationtothe first standard, itmayhappenthatzmayberemoved fromexplicitoccurrence in theequation,butthat acandyremain, andthatthenthefunctions ofpandxmaybeassociated withoneanother, andlikewise the functions ofqandy;theequationwillthentaketheform 195.]^FOBMS. 311 Weassume, asatrial solution, each oftheseequal quantities tobeequaltoanarbitraryconstant a;from the first ofthetwo equationssoobtained wehave p=e i(0,a), andfromthesecond 2= a(y,a)- Integratingboth ofthesewefindthat,bythe first, z~fi0'a)+aquantity independentofx, andthat,bythesecond, z=/ a(y,a)+aquantity independentofy. These areevidentlyincluded in,andareequivalent to,the equation *=/ t(fl,a)+/,(y, a)+6, where 6isanarbitraryconstant. This isasolution oftheoriginal equation;asitcontains twoarbitraryconstants itistheComplete Integral TheGeneralIntegral andtheSingular Integral,ifitexist, are tobededuced from thisintheusualway. Ex. 1.Solve theequation p*+q*=x+y. Theequation rearrangedintheform ^-^-(gi-y) oomes under thestandard, andwetherefore write %Pa=y-qz=a. Hence ^ 2 andtherefore which istheComplete Integral. TheGeneral Integralisgiven bytheelimination ofabetween where ^isanarbitrary function;andthere isnoSingular Integral. 312 STANDARD FORMS.f [195. Ex. 2.Solve theequations: (i) (ii) (iii) (iv)p*+q*=2x; (vr)p*-y*q=a*-y\ Eso. 3.Shew that thismethod canbeapplied tothesolution ofequations oftheform A(ft i*i)+/2(ft ,xj+/s(p,, *j)=0. Thus solvefullytheequation 196. STAITOAHD IV. Inthis class areincluded thoseequations involving partial differential coefficients, which areanalogoustotheequations included under Clairaut's form(20)inordinarydifferential equations. Fortwoindependent variablestheyarerepresented by where$isadefinite function. Asolution ofthis is z=aa;+by+<J>(a, 6), which admits ofimmediate verification. Asitcontains twoarbitrary constants itistheComplete Integral;theGeneralIntegralisto beobtained intheusualway,andthere isusually aSingular Integral Ex. 1.Solve theequations: (i)e=px+qy+pq (ii) (iii)=| (iv)zt=px+qy+ 3p ; obtaining ineach casetheSingular Integral aswellastheComplete Integral Ex. 2.Solve theequations: (i)*=A*i+iv^+ft+/(ftiA,ft>; =1 -i. (ii)*= andobtain theSingular Integralineach oaae. 197.] PEINCIPLE OPDUALITY. 313 Principle ofDuality. 197. There exists inpartial differentialequations aremarkable dualityinvirtue ofwhich eachequationisconnected withsome otherequationofthesame orderbyrelations ofaperfectlyre- ciprocalcharacter. Weshall consider hereonlyequationsofthe first order. Consideringthecase oftwoindependent variablesonly,we write asournewdependent variable Z=pac+qy z, andtherefore Wetake asournewindependentvariables pandq,which wewriteXandFforsymmetry,sothat X=pandY=q; andthenwehave _dZ_dZ _ *== ' then z=PX+QY-Z, sothattherelations between thevariables are,asstated above, reciprocal. Ifnowwehaveanequationoftheform v/r(co,y,z,p,q)=0, theabove relations transform itinto Theintegralofeither ofthesebeing known, that ofthe.other is deducible byaprocessofalgebraicalelimination. Thus leta solution ofthesecond begiven,orbederivable, intheform $(Z,X, 7)=0. Thenwehave 314 PRINCIPLE OF[197. andand The elimination ofX,F,#between these fourequationswill leave anequation inas,y,z,which willbeasolution of Ex. 1.Thesimplest exampleofanequation whioh canbetreated bythis method isthatwhich oomes under Standard IV.(196);theequation being 2=px+qy+f(p, q}, thetransformedequationisnotdifferential, butalgebraical, being infact -*-/(*, F). Thus inparticular consider thetransformedequationis "dZHence ^=dl=~2Zand==~ where e=Z Hence, eHminatingthequantities Z,F,Z,wehave which iseasily seen tobetheSingular Integral of Ex. 2.Solve theequations: (i) (ii) (iii) (iv)(px+qy-s) (p*a;+q*yfi=pq. Ex. 3.Prove thattheequations (i)^(e-px-qy,p,y)+y/ a(i-|w-qy,p,q)=/g(-^_gy ,^j,g-), (ii)F(z-px-qy,x,y~)=Q, arereducible, bytheforegoing substitutions, tostandard forms. 197.] DUMITT. 315 Ex. 4.Prove thattheequation .P,*-~Pv)+S/a(y.P,*~pz)=/(^P,*-P) isreducible toLagrange's formbychanging thevariables sothatpandyare thenewindependent variables and2-pxthenewdependent variable. Henoe solve theequation q(y-Vf+2joj7=s +xp*(x+1). Ex. 5.Solve(z-p-qy}*= 198.Theprocessofderivation ofone differentialequation. fromanother asexhibited intheprecedingarticle isreally atrans- lation intoanalysisofthegeometrical principleofduality between surfaces. When wetakeafixedquadric, which wemaydenote by 2,thenwitheverysurface $there isassociated another surface S', called itspolar reciprocal, which istheenvelopeofthepolar planes withregardto2ofpoints onthesurface S;andthesurface Sis thepolar reciprocalofSf ,beingtheenvelopeofthepolar planes withjegardto2ofpoints onS'. Thepolar reciprocalofasurfacedepends onthesubsidiary quadric, S,and isdifferent fordifferentquadrics ;thequadric mostcommonlychosen(onaccount ofthegeometrical simplicity) isasphere with itscentre attheoriginofreciprocation. Letusconsider asthesubsidiary quadricnotaspherebuta paraboloidofrevolution whoseequationis Tothetangent planeatapointAonthesurface Scorresponds apointA'onthesurface S';and tothepointAcorrespondsthe tangent planeatA'to8'.Letoc,y,z,p,qbethequantities associated withA;andX,Y,Z,P,Qthecorresponding quantities associated with A'. Thetangent planeatso,y,ztothegivensurface Sis (V,?beingcurrent coordinates) ;andthepolar planeofX,Y,Z withregardtothequadricis 316 PRINCIPLE OFDUALITY.^ [198. But,because thetwosurfaces Sand&arepolar reciprocals,these twoplanesarethesame;acomparisonoftheirequations gives Similarly, takingatangent planeatX,Y,Ztothesurface S' andnoticingthat itmust "bethepolar planeofac,y,zwithregard tothequadric, weshould obtain theequations These arethetwosetsofrelations used intheprecedingmethod. Other relations could beobtained bytakingothersubsidiary quadricsinreference towhichreciprocationshould takeplace;but theprecedingseem thesimplestthatcanbefound. 199. TheGeneralIntegralofadifferential equationinvolves anarbitraryfunction. Itmaybenecessarytoobtain aninte- gral satisfyingcertain conditions;the latter willthen beob- tained ifthearbitraryfunction berightlydetermined. The processisequivalenttothatwhich occurs inordinarydifferential equations, where thearbitraryconstants aredetermined bysome particular relation orrelations betweenspecialvalues ofthe variables. Inevery particular problemthearbitraryfunction is determined bymeans ofthespecifiedconditions. Ex. 1.Weknow thattheequation ap+bq=l implies thatthenormal tothesurfacerepresented bytheintegral equationis perpendiculartoagiven linewhose direction cosines areproportionaltoa,6,1; this isthepropertyofacylindrical surface whose JTIH isparalleltothat line. Theintegral obtained either byLagrange's method orbythemethod applied toStandard I.is where isarbitrary. Suppose that theequation ofacylinder havingits axisparallel totheline(a,&,1)andpassing through thecurve aPya=lin theplaneofxyisdesired. Thesection oftheabove surface bytheplaneof xyisobtained bywriting 2=0therein, andthus itis Accordingtotheassigned conditions itshould be x*=l+y*. Acomparison oftheseequations shews that 199.] CHABPIT'S GENERAL METHOD. 317 andtherefore also $(y-&*)-{l +(y-&*)}*. Hence theequation requiredis or,freedfrom radicals, is '(*-)-(y -&)=!. jEr. 2.Prove thattheequation representsafamilyofcones having thefixedpoint (a,6,c)forvertex. Shew thatthemember ofthefamily, whichpasaes through thecircle intheplaneofocy,hasforitsequation (a*- car)2+(&z-cy)a=(z-o)2 . Ex. 3.Obtain theintegraloftheequation p(nyma)+q(h-ma)=mx- ly, sothattheseption, bytheplaneofsty,oftherepresented surface isaconic section ofeccentricityewith itscentre ontheline General MethodofSolution. 200.Wenowproceedtoconsider amoregeneral method due partlytoLagrangeandpartlytoCharpit;itappliestothegeneral equation,whichmaybedenoted by F(*> y>*>P,3)=0, and itssuccess depends,aswillbeseen,upon theintegrationof some ordinarydifferentialequations. Ifinaddition totheforegoingrelation wehave another between thevariables andthe differential coefficients, thetwocanbe considered asapairofsimultaneousequations which, when solved, willgivepandqasexplicitfunctions ofa,yand z.Thevalues so derived, when substituted intheequation dz=pdx +qdy, willrender iteither immediately integrableorintegrableon multiplication bysome factor;andtheintegralwillbeasolution, 318 CHASPIT'S GENERAL[200. oftheoriginal equation,since thevalues ofpandqderived from ithave intheinverseprocess been obtained from thatequation. Letthen another relation between thequantities bedenotedby 3>(as,y,z,p,q)= ; ifwecanfindtheform ofO,weshallbeinapositiontousethis method ofsolution. 201.Now theintegraloftheequation givesz(and therefore alsopandq)asfunctions ofasandy;whatever these functions may be,they will,ifsubstituted intheequations F=0 and <l>=0, render them both identities. Letthen thevalues ofz,p,q(as yetunknown) besupposedsubstituted;thenthepartial differential coefficients oftheleft-hand members ofbothequations withregard toasandywill allvanish, andtherefore d_F d_F dFdp d_Fdq_ +P+ + ~ ' _n dx dzP+ dpdto+ dqdx~' d_FdFdF3p_ dFdq_ +q+ + ~ ' 9$ 30> d&dp d&dq "a~+5~9r+5"~a+o~3 =0- oyozopayoqoy f\ Eliminating~between the firstpairoftheseequations, wehave _ J___ dp dp'das)+P\dedp dpdz)+ d~x(dqdp~ dpdq)~ ' andeliminating ^-between thesecondpair,wehave _ __ dp _ _ \dydq dqdy)+q\dzdq dqdz)+ dy(ty ~dq~ dqdp)= dqd*zdp a=2~cT=o.aosdosdy oy sothatfrom the lasttwoequations, when addedtogetherasthey stand, thetermsinvolvingthesequantities disappear; andthe resultmayberearranged andwritten intheform 201.] ^METHOD OFSOLUTION. 319 dFdF\d fdF ,dF\d fdFdF\d&+p~+~+q +-p-q dpJdx which wemaylookuponasalinear differentialequationofthe firstorder todetermine <f>.Themethodapplicabletothisequa- tion istherefore theoneused inthecase ofLagrange's equation ; wewritedown theequations (189) dp_dq_dz_dot_dy_d~dF~~Q' dFd_F~dFdF~_dF_dF da}+Pdzdy+qdzPdpq dq dp dq andobtainintegralsofthese. Now inorder thatthese equations mayholdwemusthave or <J>=A, anarbitraryconstant. Ifanotherintegralcanbeobtained by equating anytwoofthe first five fractions, itmaybewritten in theform u=. Bythecorollaryin 189,u=Bisasolution ofthedifferential equation determining3>.Now <E>= isthe relation weare seekingbetweenas,y,z,p,q; andthesimplerthis relationis,the easier willbethededuction ofpandqfromO=andF= 0.We maytherefore take astherelationrequiredtheequation u=B, that is,wemaytakeanyoneintegralwhatever oftheforegoing systemofordinarydifferentialequations, providedeitherporqor bothoccur init;when thisintegralhasbeen obtained, wecombine itwithF=andcarryouttheprocessindicated inthepre- cedingarticle. 202.Thefollowing propositionisanimmediatecorollary from theprocessofthepreceding article, oritmaybeconsideredmerely asare-enunciation oftheresult there obtained : When twoequationsofthe firstorderrepresented by F(as,y,z,p,q)=0, 3>(x,y,z,p,q)=0, 320.OHAEPIT'S GENERAL[202. aresuchthatthey satisfy identicallytherelation__ _ dxdp dpda; dydq dqdy andareconsidered astwosimultaneousequations givingpandq agfunctions of&,y,and z,thenthevalues ofpandqderived from themandsubstituted intheequation dz=pda+qdy render itanexact differential. Another formmaybegiventotherelation. Let andsimilarlyfor <I>;then theequationiseasily transformed into 9$.dF _3*,3F .F-- CD--1-W-- CD-= ^+** U< Ex. 1.Solve theequation p*+ja-2pa;-2qy+%cy=0. Writing down thesubsidiary equations wehaveamong others dp___ dq_dx_dy~~ ~' Henoe dp+dq=dx+dy, sothat p-x+qy=a. Combiningthiswiththeoriginal equation, whichmaybewritten wefind Hence dz=pdx+qdy gves theintegralofwhich is 2z-b=ofl+ax+ya+ay+^j {2(a-y)a-a2 2* 202.] METHOD OFSOLUTION. 321 * which istheComplete Integral. TheGeneralIntegralisdeduoible inthe ordinary way ;there isnoSingular Integral. Theabove equation may, however, besolved withouthaving recourse to thismethod; butsome transformations and substitutions arenecessary. Taking theequationintheform wewrite Z=z-\& Jya , ,, . az .azsothat 5-=P xand^-=a-y.oxr oy*a Lettheindependent variables bechanged bytheequations and andtherefore Theequation becomes and isthus oftheform ofStandard III.;when theintegralisobtained and thenewvariables arereplaced bytheold,itwillbefound toagree withthe above. Ex. 2.Solve theequations (i) (ii) byCharpit's method. Also reduce both ofthem tooneorother oftheStandard Forms andso integrate them, shewing thattheintegrals obtained bythetwomethodsagree. 203. Intheseparticular examples Charpit's method isless laborious than theother;butthis isbynomeansalwaysthecase. Itoftenhappensthatanequationwhich furnishes aneasyexample ofthisrule isintegrablestillmoreeasily because included insome oneorother oftheforegoingStandard forms;andthiscauses the method tobelessusedthanwould otherwise bethe case. But it ismoregeneralthananyofthem, andequations integrable byany F. 21 322 CHAEPIT'S GENERAL[203. f oftheother methods areintegrable bythismethod;itismore- overimportantinthegeneral theoryasindicatingamethod of obtainingasolution- ofthe differentialequationwithoutany restrictions onitsform. The limitations tosuccess inpracticeareconnected with the integrationofthesubsidiary equations. Now theseparticular limitations arejustsuch asgiverisetothemethodsadoptedfor thedifferent Standards andreallyindicate theclassification therein adopted;infact alltheStandards areincluded inCharpit's form andintegrationispossible bythisonegeneral method whenever itis possible byanyofthespecialmethods. 204. Thus consider firstLagrange's form,which is inwhich P,Q,Rarefunctions ofOB,y,zalone anddonotinvolve porq.Inthiscase F=R-Pp-Qq, U.* WT>WKsothat -=P,--= thustwoofCharpit's equationsare dec_dy_dz P~Q"=E' theequations onwhich theintegrationofLagrange'sform de- pends. But itshould benoticed that this isnotaproofof Lagrange's method forlinear differentialequations ;theresult has already beenassumed inthederivation ofCharpit's equations. 205.Now consider thetypical equationofthe firstStandard, which is }=0, sothat F inwhich K,y,zdonotexplicitly occur;then ?*.o3^-o ?*-<> Q ''i5~~"j f-T^" oxoy fa 205.] METHOD OFSOLUTION. 323 Thesubsidiary equations noware dp_dq_das_ ~0~~0~_?~'" dp sothatwehavep=aandq=b,botharbitraryconstantsap- parently. Butaccordingtotherulewemust combine anyone integralwiththeoriginal equation, andsowehave ^(a,q)=0; andtherefore, ifq=b,wehave ^(a,6)=0. Then dz=pdoc+qdy =ados+bdy, ofwhich theintegralis z=CUB+by+c, withthelimitation between aand b. 206. Proceeding now tothetypical equationoftheSecond Standard, which is *lr(z,p,q)=0, anequationintowhich xandydonotexplicitly enter,wehave F=TJr(e,p,q), andthereforeWn-BFn5-=0,and-=0.oxay Theequationderived from the firstpairofCharpit'sfractions gives d dp_ djP~ andtherefore p=mq.Combiningthiswithty=wecanfindboth pandqinterms ofz;letthevalues bef(z)forpandtherefore mf(z)forq.Substitutingin de=pdco+qdy, wehave ^=da>+mdy,J\z) 212 324 CHABPIT'S METHOD. . [206. or whichagreeswiththeformer result. 207. Passing now totheThird Standard inwhich theequa- tion is > sothata~=oJ5~ aOCD dxopdp =_ _ dy dy' dq= dq' wehavefrom thesubsidiary equations d dx dasdp %*+%*- that is, (co,p)=a; andtherefore from theoriginal equation Solvingtheserespectivelyfor^andqwehave ^=^(5;,^; q= 3(y,a); andfollowingtherulewehave dz=6^(a,a)dcc+a(y,a)dy, theintegralofwhich is z+c= j9 t(as,o)dx+je3(y,a)dy. Ex. 3.Derive byCharpitfs method theintegral ofthe differential equationoftheformanalogoustoClairaut's form forordinary equations. Ex.4Obtain byCharpifsmethod asolution oftheequation =f(p, q), where/(p, q)isahomogeneous function ofpandqofthedegreen.1/V Solve also 208.] ,JACOBI'S GENERAL METHOD. 325 JAOOBI'S METHOD FORTHEGENERAL EQUATION WITH ANT NUMBER OFINDEPENDENT VARIABLES. 208. Ithas"been indicated in189that themethod used forthelinearpartialdifferentialequationinLagrange's form can beappliedtothecasewhen thenumber ofvariables isn;wenow proceedtoindicate themethod, due toJacobi, ofsolvingthe general partialdifferentialequation when there arenindependent variables. Thisgeneral equation mayberepresented by *(*>&&......A.a**......,<O=. where cslta;a,....... xnaretheindependentvariables andthep's arethepartialdifferential coefficients ofzwithrespecttothe SD'B. 209.We willprovethat ifinthisequationthedependent variableexplicitlyoccur (whichwillusually bethecasesince the equationisperfectly general),then theequation $=can1be replaced byanother withanewdependent variable, inwhich that dependentvariable does notexplicitlyoccur andthenumber of independentvariables isincreased byunity. The differentialequation<E>=hassome solution;letitbe represented by where/isasyetanunknown function;thenwehave du du- A o+l~Pr= OXTfa* forallvalues ofthe suffix from r=1tor=n.Letthese values ofpbesubstituted intheoriginal equation,which therefore becomes 9w 'du du fa fa fa andmaybewritten intheform * dudu. dudu 326 JAOOBI'S METHODP [209. This isapartialdifferentialequationofthe first order; the dependentvariable udoes'notexplicitlyoccur andthere aren+1 independentvariables z,ccitcca....... ,xn.Hence theproposition isproved Theintegralofthis leads totheintegraloftheoriginal equation ;itwillbeprovedtobepossibletoobtain theintegralof M*=intheform u=f(% 1,aJ8,......,a>B,z>a1}a,,....... aj, inwhich a1}aJ5......,anarearbitraryconstants. When thisintegralisknown, thecomplete integralofthe equation $=isgivenby /(0 132v......,xn,a,altat>......,an)=0, inwhich zisnow thedependentvariable and there arethe originalnindependentvariables. Foru=fistheintegralof"*&=and M?"isamodified form of <E>=0,sothatthelatter issatisfied byu=f,andtherefore ~ . (z '~df'^dz %z dz Butsince/=wehave andthereforeV which issatisfied forallthesuffixes rfromr=1tor=nhence weobtain theoriginaldifferentialequation. 210. Itisthus sufficient toconsider differentialequations from which thedependent variable isexplicitlyabsent. Ifit explicitlyoccur inanygiven equation,itcanberemoved in themannerindicated; andatransformed differentialequation 11 "- r ~, 1 7 t 210.] ^FORTHEGENERAL EQUATION. 327 -. MT=canbeobtained, theintegralofwhich willlead tothe --*- required integral. Wemaytherefore write thegeneral differential' equationintheform If,inaddition toF=0,wehave othern1equationsofthe form whereF^,F2,......,F^arefunctionsofp t,pa,......>2>B(orofsome ofthem) and itmay be,andusuallywillbe,ofxltx9,......,ccn,and where a1}aa,......,anF.larearbitrary constants, thenfrom thesen equations wecanobtain values ofpltp,t......,pnasfunctions of the as'aandthe a's.Letthese values besubstituted in then,iftheybesuch astorender thisanexact differential, the integralofitwillbethecomplete integralofF=0.For itwill beanintegral,since thevaluesop ltpat.......pnarederived from nequations,oneofwhich is.F=0; and itwillinitsexpression involve narbitrary constants, viz.theconstantsa,,aa,......,a^ andtheconstant ofintegration. Moreover theintegralisofthe form whichgivesthedependentvariableexplicitly, and therefore justifiestheassumption made astotheform oftheintegralof =0. Then1functions Fmust besuch that thevalues ofthe quantities pwillrender theforegoinganexact differentialequa- tion;andthenecessary conditions, which are forallvalues ofrand s,willserve todetermine these functions. 211. Supposethatthenequations aresolved soastogivethevalues ofpltpt,......,pnasfunctions ofthevariables x;these values will,when substituted, make each 328 JACOBl'S METHOD f[211. equationanidentity. When thissubstitution takesplaceinany twosuchequationsasFr=arandF8=aa,wehave 3a;a giving altogethernpairsofequations ;eachpairismade upof thedifferential coefficients, withregardtothesame independent variable, ofFrandFtwhen inthese thevalues ofthep'aare substituted Between the firstpairletthevalue of|jpbeelimi- oosJ nated;theresulting equationis i-+--+'+ + =o ..AJ3iLP..iJ ai...... where _ ,vJ3w3w 3du Z'i=rs-Z'i JLv>uJ' Similarlytheelimination of^from thesecondpairgives OiVy r^.j.]^r^^lfe,,......... and soon,eachpairleadingtoanequationofthisform. Now let alltheleft-hand members oftheseequationsbe addedtogether. The coefficient of^(which isequal to^ 3ov \ wiUconsist ofthesum oftwoterms, viz.theterm 211.]V, .FORTHEGENERAL EQUATION. 329 from ther'-equation, andtheterm from thes^equation ;thesum ofthese two iszero,andthus the Q term in="disappears, whatever he.thevalues ofr'and s'.The CflV resulting equationistherefore Lettheleft-hand sidebedenoted by (Fr,F g); then theequationis (Fr,F 8)=Q; and thismust besatisfied, whatever thesuffixes randsmaybe. Hence theaggregateoftheequations which these functions must satisfy mayberepresentedintheform forallvalues oftheindex ifrom i=\toi=n 1. 212. These conditions, which arenecessaryfortheintegra- bilityoftheequationdz=%pdx, mustnowbeprovedsufficient; this willbeproved byshewing that,when thefunctions Fsatisfy theforegoing equations, wehave forallvalues ofr'and s'. Thenequationsderived from thenpairsofequationscon- nected withanytwogivenfunctions FrandFtstillhold;when theyarealladded together wehave thedouble summationextendingtoallintegralvalues ofr'and sffrom 1tonbutnotincluding pairsofequalvalues since for everysuchpairofvalues theterm vanishes. Butbythenecessary conditions satisfied bythefunctions wehave 330 JACOBl'S METHOD r[212. J1.1. f *<andtherefore 2 which holds forallthevalues ofrand sgiven bythedifferent functions;andeverycombination ofthefunctions willgivesuch anequation. The totalnumber ofthese combinations is%n(n 1); andtherefore thenumber ofsuchequationsisfan(n 1). Noweachequationislinear inthequantities dp^ dp,? den,/ dtKff' which areinnumber %n(n 1)inall,that is,thesame asthe number oftheequations.Since eachright-handside iszero it follows either thateach ofthesequantities dpf/ dp,/ dfl/a' OSDfi iszero, orthatthedeterminant formed bythecoefficients ofthese quantitiesiszero. That thiscannot bethecaseappearsasfollows. LetAdenote thedeterminant. dFd_F W; 9j3j' dp,' ' dpn dp,' dp,' dp,' ' dp, then each oftheexpressions [F.v.fli S>pJ isthecomplementofasecond minor ofAandthere areinall Jw*(n 1)Bofthem;let denote thedeterminant formed by them sothat@isthedeterminant which iszerobyhypothesis. Let@'bethedeterminant formedbythecomplements inA oftheconstituents in@;thenwehave, onmultiplying and0' together, 0'= 212.] \ .FOBTHEGENERAL EQUATION. 331 Now' isnotinfinite;hence ifvanish wemust have But thiswouldimplythatamongthenequationsofthetypeF=thenquantities pcould beeliminated, thatis,that these equations would not suffice todetermine thequantities pas functions oftheindependentvariables. This iscontraryto what hasbeenassumed astotheindependenceofthefunctions F;hence isnotzero. Itfollows thateach ofthen(n 1)quantities iszero,andtherefore thattheassignedconditions aresufficient to ensure that isaperfectdifferential. 213.Wemaytherefore sumupour results, sofarobtained, asfollows : Toobtain theOomplete Integral ofanygiven equationF= we firstdetermine anintegralF1=a1oftheequation thenweobtain acommonintegralFt=aaoftheequations (Ft,F)=(Fz,FJ=0; thenacommonintegral F=aoftheequations and soon,thusobtaininginallnl newequationseach con- taining anarbitraryconstant. Thenequationswhich involve the nquantities parethensoked soastofurnishthevalues ofthep's asfunctions oftheindependentvariables and thearbitrarycon- stants, andthese values aresubstituted in Thiswhen integrated givestheComplete Integral oftheequation F=0. 332 JACOBl'S METHOD , .'' [213. Each oftheequations determining anyoneofthefunctions Frislinear inthepartialdifferential coefficients ofFr;we.have therefore toinvestigateamethod ofobtainingthecommonintegral ofasetofsimultaneous linearpartialdifferentialequations. Ex.Prove that iftheequations besolved BOastogivepltpit......,pnasfunctions of#1}xZ)...,#,2the necessary and sufficient conditions inorder that should beanexact differential arethattheaggregateofequations I \ Pi ,JP, ,+......+^5 should besatisfied forallvalues oftheindex ifromi=2 toi=n. 214. Itisconvenient toprovehereanimportant Lemma which willbeofusewhen theintegrationofthesimultaneous equationsisbeingconsidered. IfA,B}beanythree functions of2nindependentvariables 0,,/Bj,......'Mn'PiiPv......,pn,andif thefunction(B,0)bedenoted bya,andthefunction (A,a)by [A,(B,0)], then theequation [A,(B,0)]+[B,(C,A}]+[G,(A,B)]= willbeidenticallysatisfied. Consider theleft-hand member ofthisequation;itconsists ofthesum ofanumber ofterms allofthesame /orm, each of which istheproductoftwo first differential coefficients oftwoof thequantities A,B,Gandasecond differential coefficient ofthe third ofthem. Itmoreover isacyclically symmetricalfunction ofA,BandGand therefore,ifthetermsinvolvingthesecond 214.1 I FORTHEGENERAL EQUATION. 333\ . differential coefficient ofanyone function, such as0,disap- pear,alltheterms willdisappear andthus theequation willbe satisfied. Letthequantity aB9c_95a(7 dtsrdprdprdssT bedenoted byA^BC, sothatArmaybeconsidered asasymbolical operator ;wemaywrite (5,CO=(A1+A2+......+AJJK7, theoperators being obviously subjecttothedistributive law Then inaccordance with thisnotation, [A,(B,G)]=(^+\+......+AJ4CAJ andtherefore[A,(B, (T)]isthesum ofaseries ofpairsofterms forallthevalues ofrand sfrom 1toninclusive;inthecase when rand s'have thesame valueonlyasingleterm occurs for consideration. Expandingthefunctions thussymbolically represented, we findthat thetermsdepending uponthesecond differential co- efficients ofare dAdB PC9^95 9aC 9AdB9e (7dAdB d*C ^O^O Ir\ ^\ crdp,dp^os, dprdosadp$sc rdprdp, from thefirstoftheforegoing pair,and dAW_9*0_dAdB_9a(7_dAdB 9a<7 9AdB das,doordpjdp,dossdprdpjdx rdp,^aordpfix, dp,dp, from thesecond. Selectinginthesamewayfrom[B,(0,AJ]thecorresponding pairofsymbolicalterms andconsideringinthem theterms which involve second differential coefficients ofG,wefindthem tobe respectively dBdA d'OdBdA d*CdBdA 9SC95dAd^C_ ""r^\>*k <-\ *- doordpsdpfa,dairdx,dpjdp, dprdpadoc^ dp 334 JAOOBl'S METHOD ^.[214. r ) and "T"f\" f\ adprdp$x r da;,,dxrdpp, dpt Theexpression [0,(A, )]willnotcontain anysecond dif- ferential coefficients ofC. Hence in [4,(3,CO]+[,(CU)]+[(4*)] d*C the coefficient oftheterm which involves -~ isthesum of dpjp, those intheforegoing,and istherefore zero;soalsoaretheco- ***v v,- i*GVVC efficients ofthose which involve -.,,, -,^^-. opjdas toptdtKroasjdXt Ifrand shethesameweneedonlytoconsider the firstand third oftheabove lines oftermswhen inthemwewrite s=r;it willbeseen immediately that theterms in^,.--, aallJ dpr* 0p$oe rdos* vanish. Since this istruewhatever rand smay be,itfollows that all theterms involvingsecond differentials ofvanish; andtherefore, bythesymmetry,thewholeexpressionvanishes. SolutionoftheSubsidiary Equations. 215.Wenowproceedtoobtain thevalues ofF 1,Fa,......,F^ from thevarious differentialequationswhichtheymustsatisfy. TodetermineF^wehave or,what isthesamething, _ __ _ = 3^3ft3^3^30a3padp, dot,......3an3pn3pndxn Since this islinear inthedifferential coefficients ofJ^wemay obtain anintegralofitbyusingassubsidiary equations (189) 216.], FORTHEGENERAL EQUATION. 335 \ T thegeneralisedform ofLagrange'a equations. Letanyintegral ofthesystem tfcc bedenoted by where atisanarbitraryconstant;thenFl=/ t=atisanintegral oftheoriginal equation (F,FJ=0. 216.Wehavenow tofind afunctionF9such aswillsatisfy theequations Theformer ofthesebeinganequationtodetermine Ftisidentical! informwith thatwhich determines Fltand therefore weshall * have thesamesubsidiary equations ;let (ai^i.......nPi.Pai.......Pn)=constant beanintegraloftheequations (A)different from/,=a1;then (^^)=o. If$besuchafunction astosatisfy C/,.0-0, thenwemaytake Ft= <f>=a3 asthecommonintegralofthetwoequations which determine Ft. If(>donotsatisfytheequation,thenweshallhave thesubstitution of^mayberepeatedandsoonindefinitely,so thatweshallhaveaseries offunctionsgivenby 336 JACOBl'S METHOD[216. Now allthese functions $satisfytheequation when substituted forFz.Intheidentity [A,(B,0)]+[5,(G,A}]+[C,(A, letFbesubstituted forAand/Ifor5;then [a(^5)]=[c,(^/ 1)]=(aJo) andtherefore [J?T >(/1,co]=[/1,(^co], whatever maybe. First letC= ;then thisequation becomes [^(/ 1>0)]=[/ 1,(^0)]=(/ 1)0) sothat (/i.#=*!-*; isasolution of (JF.JPJ-O. ehave ",(/,, ]=[/C*7 ,&)]=(/0)=0,Next let C/=^;thenwehave sothat isalsoasolution of andsoonwiththewhole series offunctions <,each ofwhich isa solution ofthe first ofthetwoequations which determine F9,and istherefore, when equatedtoaconstant, alsoasolution ofthe subsidiary equations (A). Now thesesubsidiary equations haveonly2?i 1independent integralsattheutmost; thefunctions$,which arise from the indefinitely repeatedsubstitution in(flt^Jcannot allbein- dependentofoneanother;andtherefore iftheseries offunctions donot ceasewemustultimately come tosome onewhich is expressibleinterms ofthosealreadyfound. 217. There arethus three alternatives tobeconsidered : (1),some function<j>toftheseriesmaybeidenticallyzero; 217.]1FORTHEGENERAL EQUATION. 337 (2),some functionfaoftheseries isvariable "butexpressiblein terms oftheprecedingfunctions oftheseries; (3),some function <,ofthe series maybeadeterminate constant c. Wewillconsider these inturn. 218.(1),letfa= ;then <=aawillbethe desired integral ;foritisoneoftheseries offunctions and istherefore a solution of(F,F^)= ;also and itistherefore asolution of(FltF^)=0.Hence itisa commonintegralofthetwoequationswhich determine Ftand thereforegivesthesecond oftheequations desired, viz. 219.(2),letfabeexpressibleinterms ofthepreceding functions oftheseries;suppose where 8isadefinite functionalsymbol. Proceeding now toform wehave when thevalue offaissubstituted. But since/tisasolution oftheequations;and(f1}fj)vanishes identically,sothat thisequation becomes Buteachofthedifferential coefficients of6isafunction ofthe previouslyobtainedquantities< ;hence^>i+1issoalso. Itfollows therefore that(f>tand allthefunctions oftheseries afterfaareexpressibleinterms ofthosewhichprecede fa. F. 22 338 JACOBl'S METHOD ^ f" [219. Letusthen seek toobtain some function ofthesequantities which shallsatisfy theequations letitbegiven by * When thisvalue issubstituted theformerequationbecomes which issatisfiedidenticallysinceeveryfunction isasolution of (JP.ig-0; andthesecondequation becomes asbefore The lastequationisthus theonlyonewhich must besatisfied by ijr;and asnodifferential coefficients withregardtoFor f1occur initwemayconsider them asreplaced bytheirrespective values anda1.Any integralofthesystem oftheform 3>=aawillbeasolution oftheequationini/r ;and therefore wemaywrite ^=<3>=as, and soweshallhave therequired commonintegralofthetwo equationswhich determine Ft. 220.(3),let<,besome determinate constant cwhich wil merely depend uponthe coefficients oftheoriginaldifferentia equation;theseries offunctions thus terminates asthere isnc further function tosubstitute. Wethenproceedasinthe lastcasetofindsome function ofthepreceding quantitieswhict willbeacommon solution ofthetwoequations;let 220.] \.FOETHEGENERAL EQUATION. 339 When this issubstituted in(F,F^)=theequationisidentically satisfied;when itissubstituted in(/4,FJ=theresulting equation is,justasbefore, inwhich wemay replace fabyc.Anintegralofthis isgivenby dfa=z=d$t=i *M' which whenintegrated gives $<_," 200^=constant; andtherefore wemayasinthelastcasewrite asthecommonintegraldesired. This solution issatisfactory providedi>1. Now icannot bezero since <isdetermined asafunction of thevariables;theonlyexceptiontherefore tobeconsidered isthe case i=I,when sothat%isindependentof <f>.Now andFand/Iarereplaceable byandatrespectively;ifthen^be independentof<,itceases tobeafunction ofthevariables and there isthusnosolution common tothetwoequationstobe derived from these functions. Should thisbethe case,wereturn tothesubsidiary equa- tions (A)anddetermine anewintegraldistinct from those already obtained, which are F1=f 1=a1,^>=constant; letthisbe *fa,**, .......X.Pi.JV......,^J=constant. 222 340 JACOBl'S METHOD ^f [220. Next weperformwith thefunction S-alltheoperations which have been performedwith thefunction <;then the desired commonintegral willbeobtained, exceptinthesinglecasewhenwehave where cisadeterminate constant. From acombination oftheserespective exceptional cases, which aretheonlyones ineach ofwhich thecommonintegral Fahasnotbeen obtained, wecanconstruct acommonintegral FFor let besubstituted in(F,FJ=Q=(f1}FJ;then theseequations become =^*)8+w*>!> Now theformerequationissatisfiedidenticallysince <and arebothintegralsofthesubsidiary equations (A) ;while since and (/a thelatterequationbecomes This issatisfied by andtherefore F%=@(c'<f>-<&)=av where@isanyarbitraryfunctionalsymbol (which mayatwillbe chosen ofasimple form),isthedesiredintegral. Hence ineverycase thecommonintegraloftheequations which determine Fahasbeen found;forconvenience wemay denote itby 221.]^FORTHEGENERAL EQUATION. 341 221.Wenowproceedtoobtain.F&;itmust beacommon integraloftheequations Toobtain onewefind,bythepreceding method, anintegral common tothetwoequations which iadifferent from/ a=aa;thiswemaydenoteby *(*!......>*>Pi>Pa>......,#J=constant. Wethenform asbefore theseries offunctions then allthefunctions \ofthis series arecommonintegralsof the firsttwo oftheequationswhich determine XForinthe identity [A,(B,C)]+[B,(G,A}]+[C,(A,B)]=0, letA=FandB=/a;then since(F,fy=0,wehave And, substitutinginthesameidentityA=f tandB=feand re- memberingthat(/t,/J=0,wehave These twoequationsaresatisfied whatever Gmaybe.Now let =X;then or (^\)=(/a.<>)=0; and [/l)(/a^)]=[/a,(/ 1^)L or(/1,\)=(/a,0)=0. Thus\isacommonintegraloftheequations (Jf,^)==(/1,J'8). Similarlythesubstitution of\forGwould shew that\isa commonintegraloftheseequations; and soonthroughallthe series offunctions. Asintheformer case,thenumber ofcommonintegrals being limited, weshall intheseries come tosomeintegral\kwhich is 342 JACOBl'S METHOD. Jl[221. expressible,aswellasthose that follow it,interms ofthose which precede it,viz.,F,f l}f^,X,\,......,\_^.Thesame three alter- natives arepresented andthevalue ofFathecommonintegralin each isdetermined asbefore;either thesinglecaseoffailure is avoidedbythechoice ofanewintegraldifferent from \,orin thecase offailure ofthelatter these twocases offailure are combined soastofurnish acommonintegralThus weobtain ourthirdcommonintegral,whichmayberepresented by 222. Theremainingfunctions Ft,......,Fn_1maybederived inthesamewayastheabove;andthuswithF=weshallhave nequationstodetermine thevalues ofthep'einterms ofthe independentvariables andnIarbitrary constants, which, when substituted in dz=p 1das1 willrender itintegrable;itsintegralisthecomplete integral oftheoriginaldifferentialequation. The associatedintegralsarederivable from the results of 179,180. 223.TheforegoingisanexpositionofJacobi's method ofintegrationin itssimplest form;thereare,however, developments andsimplifications and, arising outofthese, methods ofavoidingtheexceptionalcaseswhich cannot bedealt -with here. Fortheseandforthewhole theoryofpartialdifferential equations ofthe firstorder reference should bemade tothechief authori- ties,which areJA.OOBI, "Vorlesungen uberDynamik" (Qea. Werke, Suppl.Bd, pp.248269); JAOOBI, "Novamethodus...integrandi" (Crelle,t.LX,pp.1 181) ;averyvaluable memoir byIMSOHBNBTSKY, Qrunerfs Arokiv derMothe- motik imdPhysi&,t.L.pp.278474; amemoir byGRAJNDOHGB, Memoires dalaSotittU SayaledesSciences deLiege,n8 se"rie,t.v.;andatreatise by MANSION, Thtforie desdquatiom auxdenotespourtidles,willproveofgreat use; fullreferences tooriginal authorities willbefound inthelast. Theequations (A)are,when each fraction isequatedtodt,oftheform dx,.__dF _dp,.dF m ~dt~3zv;~dt=Wr' these arethecanonicalequationsofmotion ofasystemofrigid bodies; further discussion ofthem willbefound inImschenetsky. (SeealsoRouth's Rigid Dynamics.') Wenowproceedtoconsider someexamples. 223.]* EXAMPLES. 343 4 * Ex. 1.Tosolve theequation where/does notexplicitlyinvolve theindependent variables. Wemust first transform theequationsothatthedependent variable does notexplicitly occur;letthesolution oftheequation be where theform of^hasyettobedetermined. Denoting ^!lbypan( ;idx 3-byPn+ijv?ehave andthustheequationis = *n+l*n+i-f' inwhich thedependent variable ^doesnotoccur. Hence wehave forour general formula andthesubsidiary equations give ^Pi=^P?==^2=^-^1 "-1' From thesewehave P1=1,^3=03,......,Pn= tl, whichgivenintegrals; andthenfromtheequation ^=0wehave --__ - --p>pj......jp*n+i-tnfl*n Solving-this forPn+1weshould have -P*i-xW. where^involves the .constants a;andtherefore Theintegralofthis is where aisarbitrary andmaybeassumed tobeabsorbed inthe^.Butthe integralofthegivendifferential equationis^=0;hence theintegralof s 344 EXAMPLES OF f[223. where xisgiven bytheequation Ex, 2.The casewhen/isahomogeneous function oforder\Linthe ]t?8isreadily reduced tooneoftheformsalready considered in 191. For wemaychangethedependentvariable from sto,where andtheequationisthen wherer=~^ .Theintegralofthis is provided /(o^, 03, .Ear. 3.Solve (i) (ii) (Hi)(ft- Ex. 4.Solve ^=^1+^1^2) ^3+^8 (Pi-ft)~1=0- Thesubsidiary equations are From theequalityofthe1st,2nd,4thand5thfractions wehave which whenintegrated leads to (A Wetherefore(adopting thenotation oftheprevious articles) take andwehave todetermine asolution ofthesubsidiary equations F^=a awhich Hhfl.11satisfy (J'l.J'O-o. From theequalityofthe4thand6thfractions wehave 223.1 \ JACOBI'S METHOD. 345 andtherefore wemaywrite <=p*p*=constant. Now(*i ,<W=(ft+2>a) 2ft+(ft thecontinued substitution intheequation would thusnotlead toafunction such asisrequired Wetherefore return totheoriginal subsidiary equationstoobtain anintegraldifferent from FI=O>I and$=constant;suchanone isderivable from theequalityofthe 3rd,4thand5thfractions, which give *(ft-A)" (ft-ft)' andtherefore wewrite fr=afa .Pa)i#8a=constant. Now (Flt^-(ft+AJa+tft+AX-aJ-O, and^therefore satisfies thetwoequations ;wethushave Jfa=a(ft-ft)-Ks=aa- Wenowsolve theequations F=Q, F^at, Fz=<%, tofindthevalues api,pa, p3,which are hence sothatthecomplete integralofthedifferential equationis z+A=-^log (a?t inwhich A,c^,a^arethearbitraryconstants. (Imschenetsky.) Ex. 5.Integratetheequations: (i) (ii) astf i* 346 EXAMPLES OFf[223. r tL (iii) (iv)Pi+W (vi) ftftft "ft Ithasalready been indicated(in 189 196)that several oftheforms in twoindependentvariables which admit ofimmediate integration without the useofOharpit's subsidiary equations canbegeneralised soastoinclude the caseswhere thenumber ofindependentvariables isgreater than two. Ex. 6.Inthecasewhen agiven differential equation canbewritten in theform /ifoi^aJ......>riPltPil......>Pr)=A(Xr +i,......jtfnj.Pr +11......)Pn\ thecomplete integralisthecommonintegraloftheequations where aisarbitrary. Forthesubsidiary equations are dxT_dpr from theformer wehave andtherefore /!- - bythegiven equation. Asanexampleofthiswemaytake ^ Herewemay write i~Pz)=1s where aisanarbitrary constant. Theintegraloftheformerequationis whereAandGarearbitrary constants. Theintegralofthelatter isobtainable byCharpit's method;thesubsidiary equations are dx1_-tfo?a_dpi_dpz #2+afl/jap% pj_ From thesewehave Pl+P* andtherefore 223.]"JACOBl'S METHOD. 347 R it Hence, bycombining-with theequation theintegralofwhich issought,we- have (*!~<0Pi+Pt (#2+a)=^i 5 andthesegive -a)2- (tfa+a)2 }=AI(*!-a)-fo+a), Thus andtherefore Thecomplete integraloftheoriginal equationistherefore .H.^-^j^^-tf+c^H+a+i^kzgi^, where A,A^B,aarearbitrary constants. Ex. 7.Integrate (Imschenetsky.) Simultaneous PatHalDifferential Equations*. 224. Instead oftherebeing given onlyasingle equationto determine thedependentvariable theremaybegivenanumber ofsimultaneousequations;ifthedependentvariableexplicitly occur inanyofthemtheycan allbetransformed, asin 209, sothat itshalldisappear.Theequations maythen betaken oftheform *ThistheoryisduetoBoor; seeauthorities cited in223, p.342. 348 SIMULTANEOUS PARTLLL[224. Ifmbegreaterthanntheequationscannot beindependent;for the firstnoftheequations maybesolvedalgebraicallysoasto givevalues ofthep'sinterms ofthevariables xand these, when substituted intheremaining m-n,must reduce them toidentities since there would otherwise berelations between theindependent variables. Thus ineffect theremaybegivenatmost nsimul- taneousequations ;andwemaytherefore takemeither equalto n,orlessthan n. 225. I.Let?,=??.Wehave thus nequations givingthe values ofthenquantities pinterms ofthevariables;these values, substituted in dz=p ldxl +p t mustmake itaperfectdifferential ifthegiven systemhave a common solution. Theconditions forthisarethat forallpairsofindices; andthese, asin211,lead toequationsof theform Hence thegivenfunctions mustsatisfyalltheequationsforall possiblecombinations ofthesuffixes; andthen thecommon complete integralisobtainedbytheintegrationof and ittherefore contains onearbitraryconstant. Itmayhappen however thatthefunctions Farenotindepen- dent ofoneanother;inthiscasethedeterminant A iszero,andthere willthenbeanidentical relation oftheform 225.]"DIFFERENTIAL EQUATIONS. 34-0 t But forthepin-posesofintegration Fi=f3=......=Fn= ;and thistherefore becomes <E>(0,0,...... ,0,fl^o;,,...... ,<BB)=0. Ifthisbenotanidentity,there isarelationimplied between the independent variables, which isofcourseimpossible;itthen follows thatthegiven equationsareinconsistent andthatthere isnocommonintegral.Ifitbeanidentity,thenumber ofgiven equations independentofoneanother islessthan thenumber of thequantities p,which therefore cannot bedetermined from the given equations alone; wemust therefore have recourse tothe method whichapplies whenmislessthan TO. Thus, ifthere befourindependentvariables andfourequa- tionsFi==Ft=F6=F4begiven, there canbenocommon integralinacasewhen there isarelation oftheform where there isarelation oftheform ^-fa-^.+te-'O^+fa-flO*;, there areonlythreeindependent equations. 226. II.Letmbelessthan n.Wemaysupposetheequations- reduced tosuch anumber m,thattheyareindependentofone another, eventhough theywere notsointheform inwhichthey were firstgiven.Itwillbeassumed that there isacommon in- tegralsofarasthealgebraicrelations whichgivethedependent functions interms oftheothers indicate;thiswillbethecase if these relations become identically-null equations when inthemwe make useoftheequations Fl=0,......,Fm=0. First Case. The functionsF^==......=Fmmay satisfy the equations (Fr,F,)=Q forallvalues1,2,......,mofrand s;theyaretherefore simul- taneously integrable.Todetermine thevalues ofthequantities p,other nmequations must beobtained byJacobi's method; these willinvolve nmarbitraryconstants. From theseequations andthegivenmequationsthevalues ofpmustbederived andbe substituted in dz=p 1dxl 350 SIMULTANEOUS PARTIAL* [226. theintegralofwhich isthecommon complete integralofthe original equationsandcontains nm+1arbitraryconstants. Second Case. Itmayhappenthat foroneorforseveral com- binations oftheindices intheseries 1,2,......,mwefind(Fr,F,) afunction oftheindependentvariablesonly,or(Fr,F,)adeter- minate constant. Inneither case can(Fr,Ft)bezero; the conditions that theequationsshould besimultaneously integrable arenotsatisfied andthere isnocommonintegraloftheproposed equations. Third Case. Itmayhappen that, foroneorforseveral com- binations oftheindices intheseries 1,2,....... m,wefindresults oftheform (F^F t)=f(x it^,......,&.,&,& ....... ,pj, wherefdoesnotbecomeidenticallyzerooncombination with the given equations ;letthere be Isuch combinations, sothatm+1 must notbegreaterthan n\then forcombinations other than these Itheequations (Fr,F,)= aresatisfied. "Wenowtake andsubstitute inthefunctions (Fr,F.) where either rorsatleastmustbegreaterthanm. Ifthen these functions allvanish, wehavem+Iequations which aresimultaneously integrable;andwedetermine byJacobi's method thenm lremaining equations necessarytogivethe complete integral,which will therefore contain nm I+I arbitraryconstants. Ifforanycombination (Fm_t,fk)orforone(/,,/fc)thefunction beadeterminate constant orafunction oftheindependentvari- ablesonly,then thefunctions arenotsimultaneously integrable andthere isnocommonintegral. Ifforanycombination (Fm_t,fk)orforone(ft,fk)weobtain a function </>(xlt#,......,cc^p^pg,......,pjwhich doesnotvanish invirtue oftheequations already obtained, weproceedwith the functions <f>aswedidbefore withthefunctions /Ultimately we 226.]*DIFFERENTIAL EQUATIONS. * " shall arrive atafinite number, notgreaterthan n,ofindependent equationswhich aresimultaneously integrable,andthen, inthe ordinary way,obtain thecommonintegral;orweshall obtain aresultindicating impossibilityofsimultaneous integration,in which casethere willbenocommonintegral. Ex. 1.Obtain acommonintegral (ifitexist) ofthesimultaneous equa- tions Wehave where theright-handside willnotvanish invirtue ofF1=Q=f1 z;wethere- forewrite Thus (-Fi.J^-O; also (J^ ,F3)=-2p!ft+2tf8#4=0, (Fz,Fj=2pj^ 4-Zxj3g t=; thethree equationsaretherefore compatible. LetFbetheother function required,sothat itwillbedetermined asacommon integraloftheequations (^,^ 3)=0=(^,^)=(J'4JF1); consideringitasanintegralof (^,^=0, wewritedown theequations dx^ dxz_dx^_dx_dp1_dp%__djoa__^p4_ ~ls^= ~l^~ls z~ -tf4~~p~i Pa~ Pa~ P*' oneintegralofthese is p1=oxa, where aisarbitrary ;wetherefore tentativelywrite JP4-&-a.*J73 Wethen find (^4,FJ=0; andC^^i)-^-^*S*3 Nowonsolvingtheequations ^1=0=^3=^3; F4=a, 1 1wefind =ax, =-x P3=^i, Pt=-&2, 362 SIMULTANEOUS EQUATIONS."[226. andtherefore p^p t=x^cz, sothat (Ft,F3)=0. Henoe wehave thecommon solution intheform Pi=ax B. Toobtain thecomplete common integral wehave dz=a (#9cfa?i+Xjda! a)+- ( Ut andtherefore thecommon integralis where aaridbarearbitraryconstants. Ex. 2.Obtain other integralsofthepreceding equationsintheform (i)=a#1a;4+- CL (ii)a=2 (iii) Ex. 3.Obtain common complete integralsofthesimultaneous equations: Lp&j+3^4) jp4+(a?a+^4~3^1)^3=01 . ^t+$1-^z)pt+(#8^1~x*)Pa=OJ' +Jf^-Ol -ar^^=Oj' (Imschenetsky andGraindorge.) MISOELLANEOCJS TETAM 1.Integrate theequations: (i) { (ii)p (iii)^2(y- 2.Form thedifferential equation whose complete integralis where a2+^s+ya=oa ,abeing agiven constant anda,/9,-yotherwisearbitrary. I^rom thedifferential equation form thesingular integral Illustrate theconnection ofthecomplete, general andsingular integrals byageometrical interpretationofeach. " MISCELLANEOUS EXAMPLES. 353 S 3.Integrate andfindtheequationofthecone oftheseconddegree which satisfies this equation andpasses through thepoint (1,2,3). 4.Integratetheequation where X,F,Zarethesame quadratic functions ofx,y,zrespectively. Integratealsowhen theyarequartic functions;alsowhen they aresextic functions. (Eiohelot.) 5.Prove that if then andhence that Shew alsothat *P(* Similarly prove that a.,.1{xe~lea>}= 6.Solve theequation (Z1- where-^-^ 1*1+** 2^2+^, 3' (Hesse;) andintegrate theequation (Sohlafli.) 7.Solve theequations: (i). (ii) (iii) (iv) 8.Find theequation ofasurface whichbelongsatonce tosurfaces of revolution defined bytheequation pyqx=Q,andtoconical surfaces denned bytheequation px+qy=z. F. 23 354 MISCELLANEOUS 9.Ifa=/(#, y)beanysolution oftheequation then thecurves represented bytheequation areanorthogonal system such that theproduct ofthecurvatures atai pointisconstant. y}donotcontainy,theform ofthefunction isdetermined by where e#=28^(2*sin ^6)-^.' ^'(^am-, FandEbeingthefirstandsecondelliptic integrals andthemodulus ineai casebeing2~*- 10.Find thesurface which outs atright anglesallthesphereswhic passthroughagiven point andhave their centres onagivenLinepassii through thatpoint. 11.Find thesurface inwhich thecoordinates ofthepointwhere tl normal meets theplane ofxyareproportional tothecorrespondingcoord nates ofthesurface. 12.Find thesystemofsurfaces orthogonaltothecurves cosh a; :coshy:coshz=a :b :o. 13.Prove thatasolution ofthedifferentialequation 9tt30 "dw dx3v 9.8~~ fa wheretf>and<//arearbitraryfunctions ofx,yand s. Prove alsothat this isthegeneral solution. 14.Shew that, ifthesimultaneousequations 'du fa du haveasolution different fromu=constant, then (YZ'-YZ)dx+(ZX'-Z'X)dy+(X7'-X'7)dz=Q isreducible toanexactequation, from theintegralofwhich suchcommo solution maybederived. EXAMPLES. 355 Have theequations du ,du .du _ -5-4-#y-5-=0, dy^Oz' acommon solution other thanu=constant? 15.SolvebyJaoobi's method theequation (Imsohenetsky.) Shew thatbygeneralisationoftheformulca, which inthecase oftwo independent variables aretheanalytical expressionoftheprincipleofduality, thisequation canbetransformed intoonewhich islinear inthepartial differential coefficients ofthenew variable;andhenceintegrate theabove equation. 16.SolvebyJacobi's method (Amp&re, andGraindorge ;) alsosolve theequation i=ft (Pi+^a)+-riJ32(ft+*aft)- (Imschenetsky.) 17.Obtain thecomplete common integralofthesimultaneous equations: (Collet.) 18.Obtain thecomplete common integral of (V-V)^i-(*I*B~*Wt)P*+ (*a*s-^1^4)A"01 (a-arB2)^a+(^2^3-^1^4)^8+(*i*a~^4)^4=OJ' ^.ndthat of (Collet.) 232 CHAPTER X, PARTIAL DIFFERENTIAL EQUATIONS OFTHESECOND AND HIGHER ORDERS. 227. ITwillbeassumedthrough practicallythewhole ofthis chapterthatthere areonlytwoindependent variables; thenotation alreadyused forthepartialdifferential coefficients ofthe first order willberetained, and itwillbeconvenient tointroduce similar symbols r,8,ttorepresentthose ofthesecond order, which are thusdenned : Anequationissaidtobeofthesecond orderwhen itincludes oneatleast ofthese differential coefficients r,s,tbutnone of ahigherorder;thequantities pandqmayalsoenter intothe equation,thegeneral form ofwhich willtherefore be F(,y, z,p, q,r,M)=0. Thecomplete integraloftheequationisthemostgeneral relationpossible between#,y,zsuchthat,when thevalue ofz derived from itandtheassociated differential coefficients thence formed aresubstituted inthedifferentialequation, thelatter be- comes anidentity. Nocondition isannexed tothedefinition in regardtotheform ofthecomplete integral, whichmayinvolve inits expressioneitherarbitrary constants orarbitrary functions orboth. Anintermediary integralisarelation intheform ofapartial differentialequationofthe firstorder such thatthegivendiffer- entialequation canbededuced from it.Itdoesnotnecessarilyexist 227.]* EQUATIONSOFTHESECOND ORDER. 357 aaonedistinct from,andderivableimmediately bymere differen- tiation of,thecomplete integral ;when suchanintegral, however, hasbeenobtained theapplicationofthemethod ofthepreceding chapterwillgiveanintegralwhich mayactually be,ormay only beaparticularcase of,thecomplete integral. 228. Hitherto ithasbeenpossible onlyinparticularcases to integratethegeneral equation. Themoatimportantofthese cases isthat inwhich the differential coefficients ofthesecond order occuronlyinthe firstdegree,sothattheequationislinear; itsmostgeneralform isthen inwhich R,S,T,Varefunctions ofon,y,z,pandq.This equationwillnowbediscussed; butbeforegivingthemethods which havebeenused forits'integrationitisdesirable toconsider somespecialforms which aresimple andcanbesolvedimmediately; itwillthenbepossibletoexclude these cases afterwards from the generaldiscussion. One ofthesimplestcases is r=/(#), "bz r sothat^-=I/()das+^>(y}, where</>isanarbitraryfunction ;anotherintegration gives where both$and^arearbitrary. Ex. Integrates=constant. Similarly wemay integrate r+Mp=N, whereMandNarefunctions ofxandofyrespectively;itmay bewritten ybeingconstant forpurposesofdifferentiation andintegration withregardtoas ;andthus 358 MONGE'Sr [228.cr where isanarbitraryfunction;andtherefore z=jdxe~IMda> [IV**Ndx+< (y)]+ i/rheinganarbitraryfunction. Ex. Integrate (i)8+Mp=N; (ii) Mangesmethodofintegration oftheequation 229. Monge'smethod consists inacertainprocessforthe discoveryofeither oneortwointermediary integralsoftheform u=f(v) whereuandvarefunctions of#,y,z,p,q,and/issomearbitrary functional symbol;there isthusimpliedinthemethod atacit assumptionthat the differentialequation admits ofsuch an integral.Itistherefore inthe firstplace propertoenquire whether thisassumptionisjustifiableinthegeneralcase and, if itshouldprovenottobeso,toindicate howthegeneral equation mustbelimited sothattheassumption maybe fairlymade; forthis purposeitwillbesufficient toproceedfrom thesupposedinter- mediary integralandobtain thecorrespondingdifferentialequa- tion. 230. Since y=f(v) anduandvarefunctions ofas,y,z,p,q, wehave du du "du dudffdv dv 3u 3y . oy*dzop dqdv\dy*dzdp dqJ iJf Eliminatingthequantity-^between these twoequations wefind, astheequivalentdifferentialequationfreed from thearbitrary function, Ul(rt-^=^............(1), 230.]' EQUATION. 359 1 whereE T,$,,TI}UitT^aregiven bytherelations P,y \p thesymbols [-1), denoting, asusual, -=-=*-*- > J \ec,y/dssdydySOD Tfthen this differentialequationofthesecond order bethe same astheoriginal equation wemusthave , JRtStT,V.andJ '*=-:= 1 ana^~STF' which arefourequationsinall.Nowwhen Rr+Ss+Tt=V (2) islooked uponastheequationtobesolved, these fourequations justobtained willbeequationssatisfied bythequantities uandv fromwhich theintermediary integralof(2)maybeconstructed. Butonlytwoequationsarenecessarytodetermine asfunctions of theirindependentvariables thedependentvariables uandv;they maybetherefore considered asgiven byanytwooftheequations though,inpractice,these might provetoodifficult tosolve.When these values aresubstituted intheremainingtwoequationsthe latter mustbecome identities;andtheywillinthisstate involve thefunctions R,S,TandVoftheoriginaldifferential equation. There will thus betworelations amongthesefunctions of oc,y, z,p,qwhich must beidentically satisfiedinorder that thedifferen- tialequation (2)mayhaveanintermediary integral oftheform 360 MONGE'S __" [231. 231. There isanimportantdeduction from this tobenoted, thoughnotaffectingourpresentaim;itwould beuseless toseek anintegraloftheassumed intermediary form foranydifferential equation which isnotoftheform And, justasintheparticularcasewhenU= 0,which hasbeen already considered,itmaybeprovedthatadifferentialequation ofthisformcanhaveanintermediary integraloftheproposed type onlywhen twoidentical relations amongthecoefficients R,S,T> V,Fare satisfied. Ex.When there arethree independent variables, thesemaybecon- veniently denoted bya^,xz,xaandthecorrespondingdifferential coefficients of2bypl}jtjjj,pa.Prove that,ifeveryfirstminor ofthedeterminant 80 80 (0,fa#being functions of,o^,#a,xs,p1,p2,p3)vanish, then theequation whereFisanarbitrary function,willlead toadifferentialequation ofthe second order oftheform whereB^,Ry, ...,R^,Tare functions ofthevariables andthefirst differen- tialcoefficients ofzonly,andthatthecoefficients Rsatisfy therelation Information onthis class ofequationswillbefound inEuler, Inst. Colo. Int.,t.iii.p.448,andLegendre, M&noirea doFAcocMme desSciences, 1787, p.323. 232. Ittherefore follows thatwemayconsider asthemost generalcasetheequation Rr+Ss +Tt+U(rt-8^=V; thelinearequationisincluded inthis,being given bythepar- ticular casewhen 17= 0. >32.1 EQUATION. 361Jt We-nowassume tJiat therelations between thequantities R,S,T, 7andVnecessary forthepossession ofanintermediary integral of heassumed formaresatisfied,andweproceedtodeduce this ategral. Wehave always dp=rdcs+sdy, dq=sdai+tdy ; rhen.wesubstitute intheabovegeneral equation thevalues ofr ndtderived from theseequationsittakes theform Idpdy+Tdqdx+Udpdq Vdxdy =s(Rdy*-Sdtcdy+Tdx*+Udpdx+Udqdy). ^ow let u=aandV=b where aand barearbitrary constants) betwointegralsofthe jquations ftdpdy +Tdqdx+Udpdq Vdxdy=0, Rdy*+Tda?+Udpdx+Udqdy=Sdcody, dz=pdx+qdy, tand vbeingtherefore functions ofx,y,z,pandq. Hence wehave du .du , . , , ,dv iad + vhich must beequivalenttotheequationsofwhich u=aand i=b aretheintegrals. Nowsolvingthese fordpanddq,and isingthesymbolsof230,wefind indtherefore -Uflpda- U.dqdy=T^da?+RJiy* ,\fu,v\ /u,v\ fu>,v\ fu,v\ }jj+J-i- )+[g+M-l +H-\p\dndy\\y,qj Uff/M\p/ \p*/ J =T,da?+Rjly*-Sfody; andsimilarly weobtain (U^dp+T.dai)(U 1dq+R 1dy)=(U lVl+R 1TJdxdy, orRjlpdy+T^dqdx+U^dpdq-Vjixdy=0. 362 MONGE'S" [232.*r Thesebeingidentical with theformerequations, wehave -R 1_T,_^_Ft_^lR~T~U~V~S' andtherefore theequationtobesolved becomes Rj-+S,a+Tj+Z7;(rt-sa )=V l. Butwealready know thesolution ofthisequationbecause itwas derived fromanintermediary integral ;andthisintegralis u=f(v), which istherefore anintermediary integralasrequired. Wethus derive theintegral bymakingoneofthefunctions deduced from thetwosubsidiary equationsanarbitraryfunction oftheother. 233. Letusconsider inparticularthe case ofthe linear equation whenU= ;thesubsidiary equationsarenow Rdy*+Tdx*-Sdxdy=0, Rdpdy+Tdqdx=Vdxdy. Astheformer ofthese isoftheseconddegreeitcan,ingeneral, beresolved intotwo-distinctequationsofthefirstdegree. Since thenecessaryconditions fortheexistence ofaninter- mediary integralaresupposedtobesatisfied, itfollows thatone atleast oftheequationsofthe firstdegree will,when combined with Rdpdy+Tdqdx=Vdxdy andwith dz=pdas+qdyifnecessary,lead toanintegral system which determines uand-u;andthere willthusbeobtained an intermediary integraloftheform u=/(). And itmayhappen that each ofthetwoequationsofthe first degree similarlytreatedwill^eadtointegral systemsofthedesired form :andthere willthenbeobtained twointermediary integrals i=/(i). ,=0(w,)- Tf8'=4iRT there willbeonlyasingle equationofthe first degree equivalentto Rdy*+Tdx*-Sdxdy= ; thissingle equation will,since thenecessaryconditions aresatisfied, lead,byasimilarprocess,toanintermediary integral. 234.]*EQUATION. 363 234. Passing nowtothemoregeneralcase in.whichUisnot zero,wemay similarly provethat oneintermediary integral will, andtwointermediary integrals may be,derivable fromthesubsidiary equations, providedtheconditionsnecessaryfortheexistence ofan intermediary integralare satisfied. Letthesubsidiary equation which involves Vbemultiplied byaquantity X,asyetindeter- minate, andadded totheother;theresult is Rdy*+Tda?-(S+\7)decdy+Udpda+Udqdy +\Rdp dy+\Tdqdx +\Udpdq=0. Now thiscanberesolved intotwolinear factors soastobe equivalentto (Rdy+kTdoo+mUdp\(cfy+^da?+- cZgJ=0, providedthequantities k,m,\besuch astomake thecoefficients oftheseveral terms intheexpanded productthesame asbefore. Applyingthiscondition wefind thattherelations tobesatisfied bythesequantitiesare kT-=\T, mU=--\R, rU=U '> tfl K R these areallsatisfied bym=k=\jj., provided \bedetermined bytheequation X"(RT+UV)+\US+U* =0. Letthetwovalues of\furnished bythisequationbe\and which willbeunequal except when thetwo subsidiary equations maybereplaced bythetwo equationseach resoluble into linear factors when thevalues of k,m,Xaretherein substituted, which twoequations,afteraslight reduction, maybewritten : (Udy+\Tdon+\Udp) (Udx+\Rdy+\Udq)=0, 364 MONGB'SP [234.ru Toobtain thefunctions uandv,fromwhich anintermediary integral maybeconstructed, wemust combine inpairsafactor from the firstwith afactor from thesecond. But ofthefour possible combinations twomustbeexcluded, viz.,thatobtainedby combiningthe first factors intheseequations,foritwould lead to aresult whichobviously would notfurnishanysolution: andthatobtained bycombiningthesecond factors intheseequations,foritwould lead toaresult Udx=0, whichobviouslyalsowould furnish nosolution. Hence theequa- tionsmayagainbereplaced bythetwopairsofequations Udy+\Tdx+\Udp=Q\ Udas+\Rdy+\JJdq= J' and Udx+\Rdy+\TJdq= } ffdy+\TdjE+\Z7cZp=j' From oneofthepairsweshallhavetwointegralsoftheform u=aandv=b;andtherefore alsothroughthatpairweobtain an intermediary integral And itmayhappen,asinthesimplercase of233,thatwe canobtain anintermediary integral througheach ofthepairsof equationsofthe firstdegree. These twointegrals, whichmaybedenoted asbefore by ^=/K)>w9=^(v a), areintermediary integralsoftheoriginaldifferentialequation, and aredistinctexcept when 8*=4,(RT+UV), when there isonlyasingle intermediary integralobtainable. 235.Wemaynowproceedfurther intheintegrationfor either thelinearequationof233orthemoregeneral form of 234.Takingtheintermediary integralobtained ifthere beonly one,oreither oftheintermediary integralsifthere betwo,wehave adifferentialequationofthe first order; thecomplete integral (and theassociatedintegrals)ofthiscanbeobtained bythemethods of Chap.ix.Thisintegralwillbethefinalintegraloftheoriginal equation. 2HG.] EQUATION. 365 23G. Inthecasewhen thure aretwointermediary integrate womayapplyanimportant proposition (nowtobeproved) which willconsiderablyshorten thefurther labour ofderivingthis final integral.Thisproposition maybeenunciated asfollows : Whenwehave obtained twointermediary integrals oftheform andweconsider them assimultaneousequationstodetermine pand <lasfunctions nf#,//,and z,thevaluesofpandqgiven bythese etjtwtiunswillbesuoh uatorender dz=pdtf+qdy inter/ruble. Assumingthisproposition established wehave thereforemerely tosolvo thotwointermediary integralsaasimultaneousequations inpandq;tosubstituto thevalues ofpandqthence derived in dz=pdx+qdy andink-grate.Thuresult willbothefinalintegral. 237.WunowprouuudLoestablish thepropositionenunciated above. IMiF~ and <I>=respectivelydenote theseintegrals, mithat.F=u 1-f(v 1),<I>ut-f(v t),and first let .F=bea solution fthuequation sa~V. Wiihavu onlythesingle equation ^=0, which isnotsufl&cient touiiablo ustooxproHH r,8and teach asfunctions ofx,y,z,pandq; wocanoxproBB anytwoofthorn interms ofthethird andof iluantitioH uxplicitly indepcindontofthem.When these values are .sul)Hl,it.utud intho difli-niiitialoquation,thelatter willcontain one scjtofUsnna involvingthinsecond differential coefficient ofthe duplmdcntvariably andanother setnotinvolving it;andthe (.(liiuLionistobesatiafiud identicallywithoutregardtothis differential coefficient. Now sinceF= 0,wehave nV+Ts+5~02Jdp oq 366 MONGE'S* [237. * f dF dF dF 7\Fwhen forbrevity wereplace =-+p=-byFxand=-+q=-byFu, (7 ^ thesegive dF dF =-=-x--v. dq dpv Letthese values ofrand tbesubstituted inthedifferential equation ;itbecomes dq Thismust besatisfiedidentically withoutregardtos;and therefore thecoefficient ofaandthetermindependent ofitmust both vanish. Ifthiswere not so,tfceequation would determine 5 (andtherefore alsorandt)asfunctions ofx,y,z,pandq_a result which, asweknow, cannot bededuced from thesingle equation F=Q. Hence wehave Thesameequationswillbesatisfied whenwereplaceFby <3>;andwemaytherefore consider Fand <3>asthesolutions ofthe equations -. f " oq'dp dpdq"* f+Tff\'_u& _u&=0 . dpdq \dp/* dq "dp 238.Wemustnowconsider twocases. (1)The linearequation, whenU= ;let andfabethe roots of 238.]'EQUATION. 367 sothat thesecondequationbecomes '30 fc3@\/3@ fcc~ ,__ Wemaytherefore write 3-?-f3?_0 3?^a/j' 9<X> tKE> 32^s3p' thus associating twith^and ,with <l>.The firstequation,on 3 dividingoutby^~,becomes andtherefore R%JF a+TFU+V= 0. ButT=R^ 9,andthelastmaytherefore bewritten Similarly *,+^+ =0. From thelasttwowehave r?*_??.* a?_eF?*. andthorefore ,-*. which isthecondition(202)tobesatisfied bythetwofunctions Fand$inorder thatthevalues ofpandqderived fromF==<3> assimultaneous equationsshould render dz=&c+gdy intograblo.ThisprovesthepropositionforthecaseofU= 0. (2)ThegeneralformwhenUisnotzero. Wenowproceed exactlyasin234;the firstequationin is multiplied byaquantity Xgiven by 368 MONGE'S- [238. e ^ and isadded tothesecond; theresulting equationisresolved into factors foreach ofthevalues ofXandthelinear factors are combined asbefore, givingtwopairsthatmayberetained. These are, if\and\bethetwo roots, and From the firstandthird oftheseequations wehave _ = _" dpa dp~ Xj,dp"dq\dqdp' andfromthesecond andfourth F^_<I> f^-L???*.!. !????. *a-*a-^a-q?+^dqty5 andtherefore Ftt 4? ffl HF<=0. Thisshews that, forthemoregeneralform oftheequation whenF=Q=3?aretreated assimultaneousequations,thevalues ofpandqthence derived aresuch astorender dz=pdx+qdy integrable. Hence thepropositionisprovedingeneral. When these values ofpandqaresubstituted, theintegraloftheresulting equationisthefinalintegraloftheproposeddifferentialequation ; itwillinvolve initsexpressioneitherimplicitlyorexplicitlythe twoarbitraryfunctions which occur inthetwointermediary integrals. 239. Thestatement ofthemethod ofsolution, asderived from thepreceding investigation,iscontained inthefollowing Rules. ;39.] EQUATION. 369 * * RULE I.When theequation Rr+Ss+Tt=V 3integrable bythis rule,wetransform itbytheequations dp=rdx+sdy, dq=ado:+tdy, ntoRdpdy+Tdqdx-Vdocdy=s(Rdy* -Sdacdy +Tda?) ; veresolve Rdy*-Sdxdy +Tdtf= ntothetwo dy ^dec=0,dy^dx=0. Prom oneofthese linearequations andfrom theequation Rdpdy+Tdqdx-Vdocdy=0, combined ifnecessarywithdz=pdx+qdy, wecanobtain two ntegralsM t=aitv 1=bt;then Mi=/iW where/isanarbitrary function, isanintermediary integral From theother linearequation,combined withthesameequations, wemaybeable toobtain anotherpairofintegralsita=aa,va=&a; inthat case,ua=/ 8(fla)is,anotherintermediary integral, /abeing arbitrary. Todeduce the finalintegral weintegratetheintermediary integral,ifonlyonehasbeen obtainable, bythemethods which applytodifferential equationsofthe first order. Ifthere betwo intermediary integrals,wesolvethem asequations giving pandq andsubstitute in dz=pdas +qdy, whichwhenintegrated givesthecomplete integral. RULE II.When theequation Rr+ Ss+Tt+U(rt-sa )=V isintegrable bythis rule,weeither canobtain twointegrals w,=a^ andt=6toftheequations Udy+\Tdx+\Udp= Udx+\Rdy +\Udq= orcanobtain twointegralswa=aaandv3=&aof Udx+\Rdy+\Udq=Q\ Udy+\Tdtc+\Udp=0}' F. 24 370 MONGE'S r[2 f T where\and\aretheroots of i orwemaybeabletoobtain both setsofintegrals. Thenu^=/j(vjandu3=/ s(fla),where/xandftarearbitral jareintermediary integralsintherespectivecases.Weprocei I from theseexactlyasinRule I. [ ! 240. Itmay,however, provenottobepossibletoobtain, fro : thetwointermediary integrals,values ofpandqsuitable f jinsertion in dz=pdx+qdy ; andinthat casewemayproceedtoobtain the finalintegralI ; integratingoneoftheintermediary integrals, adoptingforth ! purpose Charpit's method asindicated in 201. But withoi i actually going throughtheworknecessaryinthatmethod toderr 1theadditional relation betweenp,qandthevariables, itwill 1 1sufficient totake, asthis additional relation, anyparticularfir, 1 integralofthegeneral systemother than thatwhich isbeir ! directly integrated;thuswemaytake where aisanarbitraryconstant. Since anarbitrary constant is particularcase ofanarbitraryfunction thevalues ofpand derived from theseequationswillbesuch astorender dz*=pdx+qdy integrable ;andtheintegralwillinvolve onearbitrary function andtwoarbitrary constants, viz.,aandtheconstant ofintegratioi This result constitutes thecomplete integraloftheintermedia! integral ;thegeneral integral maybederived byLagrange'snil (180),byconvertingone ofthearbitrary constants into a arbitraryfunction oftheother andeliminatingthisremainin constant between theequation sotransformed andthatdeduce from itbydifferentiation withrespecttothatconstant. 241. Thismethod, however, ceases tobeeffective inthecas inwhich theroots ofthequadratic inXareequal ;there isthe onlyonesystemofintegrals given by^=aandv1=b,andsothei isonlyoneintermediary integral given by 241.]* EQUATION. 371 " . andthismustbeintegrated.Just asbefore wemayavoid theuse ofthegeneral method fortheintegrationofanequationofthe first orderbycombiningthegeneralandparticularfirstintegrals u1=f(v i]and vi=b. Thevalues ofpandqhence derived willevidently satisfythe condition of202,andtherefore when substituted intheequation dz=pdas+qdy willgiveanotherintegraloftheform wl=c. Ifpandqoccur inw1}theymaybeeliminatedbymeans ofthe formerequationsvl=bandui=/(&)Isothat M!=c isacomplete integraloftheequationsince itinvolves twoarbitrary constants 6and c.Toobtain thegeneral integral wemustmake canarbitraryfunction of6andeliminate 6between theresulting equation andthat derived from itbydifferentiation withrespect to6. Thus inthecases,when theroots ofthequadraticareunequal andwhentheyareequal, weareledtoageneral integral,intothe expressionofwhich twoarbitraryfunctions enter. Itmaybenoticed that theforegoing reasoningwouldapply equally,ifthere hadbeentaken instead oftheparticular integral a=a* some otherparticular integralsuch as (kand Ibeing disposable constants).Thisparticular integral may, infact,betaken soastorender thesubsequent integrationas easyaspossible. Exampleswillbefound below. Ex. 1.Solve r=a?t. Substitutingforrand tinterms ofawehave sothatthesubsidiary equationsare -a?d3?=Q, <fldxdq=Q.242 372 EXAMPLES OFr[241. " r Theformer canberesolved intothetwo dyadx=Q, therespective integrals ofwhich are Taking thefirstoftheseandcombiningitwiththesecond ofthesubsidiary equations wefindthatthelatter becomes dp-adq=Q, which, whenintegrated, gives p-aq=A'. Hence oneintermediary integralis p-aq=<l> 1(y-ax). Taking thesecond equation y+ax=B, andproceedinginthesame way, wefind which leads to andtherefore asecondintermediary integralis Wenow, inaccordance withourrule, treat these assimultaneousequations givingthevalues ofpandq;andwefind dz=\dx {02(y+ax]+X(y-ax)}+dy{<3(y+ax}- t(y-ax}} ~ 2a which canbeintegrated. Let <f>(t)=^jcf>,(t)dtand then theintegralis e= (j>(y+aai)+^(y-ax). Thearbitraryconstant ofintegration maybeconsidered asabsorbed iu either ofthefunctions<pand-^.Since:and <aarearbitrary, and&are alsoarbitrary. Ex. 2.Solve Transformingthisbytheusual relations wefindthatthesubsidiary equa- tions are a+cp)dxdy+ (a 241.]* ^MONGE'S EQUATION. 373 Theformer ofthese gives onlyasingle equation (6+eg)dy+(a+op]dx=0, sothatonlyasingle intermediary integral canbeobtained fortheequation, assumedintegrable bythismethod. When this iscombined with dz=pdx+qdy, itgives adx+bdy+cdz=0, sothatoneintegralofthesubsidiary equationsis ax+by +cz=A. Eliminatingtheratiody:dxbetween thesecond subsidiary equation and themodified form ofthefirstwehave (6+eg)dp=(a+cp)dq, theintegralofwhich is Bbeinganarbitraryconstant. Hence theintermediary integralis a+cp=(b+cq)<(ax+by+oz). Thismustnowbeintegrated jLagrange's processforlinear equations maybeadopted. Denoting (f>(ax+by+ci) by<,wehave astheauxiliary equations dx dy_dz cc<p b<j>a From thesewehave adjs+bdy+odz=0, sothat ax+by+cz=C, aud$=$(ax+by+&)=$((7)isaconstant. Hence forasecondintegral The finalintegralofthedifferential equation iaJJierefore y+x<j) (ax+by+cz)=-^ (ax+by+cz), where and-fyarearbitraryfunctions. Itmayalsobeexhibited intheform *=xB(ax1-by+cz)+yx(ax+by+cz), where Qandxarearbitraryfunctions. Ex. 3.Integrate (i)r+ka?t=Za*, (1)when kisnotunity, (2)when kisunity; (ii)afir+2xys+y*t=Q; (iii) <fr-Zpgs+pit=Q; (iv)x3r-ytt=Q; (v)r- r 3*74 EXAMPLES OFr r [24" Ex, 4.Integrate theequation ar+bs+ct+e (rtsz")=h, a,b,o,e,h being constants. Theequation inXis or,ifwewriteXwi+e=0, theequation which determines mis let7?^andTTIJbeitsroots. The firstsystemofintegralsis adx+edp-m^dy=0| =Q)'ady+edg- sothatoneintermediary integralis ex+ep-m^y=F(ay+aq-rn^x). Thesecond systemofintegralsis ady+edqm^dx=0, cdo;+edp mtffy=0, andtherefore asecond intermediary integralwould be cx+epm$=& (ay+eq m^x). Ifitwere possibletosolve these intermediary equationssoastoexpress andqinterms ofxandy,thefinalintegralwould beatonce derivable;bu thisnotbe'ing thecasewecombine anyparticular integralofthesecond wit thegeneral integralofthefirstsystem.Thuswemaytake andthen F(ay+eq-m s!c)=(m t-ini)y+a, sothat,ifbetheinverse function ofFandtherefore anarbitraryfunotioi wehave ay+eq*=m^s+*{(m?-m^)y+a}. Thus edz=-<& theintegralofwhich is ee+lca;z whereQifianarbitraryfunction(sinceitisgivenby (mz-ml}Q(z'}=^(e)dz, and*isarbitrary)and|3isanarbitraryconstant. This istheComplete Integral ;toobtain theGeneral Integral weeliminat abetween theequations ez+J(oc2+ay*)=m.<py+ax+6{(m^-m^y+a}+x(a)\ Xdenoting anarbitraryfunction. -241.1" MONQE'S EQUATION. 375'J ! Ex. 5.Solve (i)#-rt=a*; (ii)qr+(p+x)8+yt=-q+y(&*-rt); (iii) Zpqyr+(p*y+qx)a+xpt=p*q (rt-a2 )+xy. Ex. 6.Solve Theequationwhich determines mis"j^j-*^*""" m?+Zpqzm+pzqW=0, t sothat thetwovalues ofmareequal,thecommon value being-pqz ;and thesystemofintegralsreduces toonegivenby z(1+g*) dy+e*dq+pqzds;=0. Theformer bymeans of ds=pdx+qdy gives,after division by2, daa+pdz-\- edp=Q, theintegralofwhich ia thesecond similarlyleads to dy+qdz+zdq=Q, theintegral ofwhich is y+qe=b, sothattheintermediary integralis whereFisarbitrary. Proceedingasindicated in241,wehave a?+pz=a, andtherefore zdz=pzdx+qzdy =(a theintegralofwhich is Ageneral integralisfound,asthere explained, byeliminatingcbetween theequations and {x-$(c)} <t>'(o)+{y-+(c)}V(c)+c=0, \lrand<f>being arbitraryfunctions. 376 PRINCIPLE OPr [241.f rL Ex. 7.Solve (i) xqr+ypt+xy (a3-rt)=pq; (ii) q*r+4pq8+p*t+p*q* (rt~a2 )=a3 ; (iii) Ex. 8.Prove theconverse oftheforegoing general result, viz.,Letthe equationofasurface be $(a;,y,z,a, b,c)=0, where a,6,careconnected byanytwoconditions oftheform x(a,I,c)=Q=+(a, b,o); shew thattheequationofitsenvelopewillsatisfyapartialdifferentialequa- tionoftheform Kr+S8+Tt+l7(rt-8*)=V, thecoefficients ofwhich satisfy therelation Principle ofDuality. 242. Thisprinciple,which wasshewn(197)tobeeffective indeducingfrom thesolution ofoneequationofthe firstorder that ofanother associated with theformer byrelations ofaper- fectly reciprocal character, maybeappliedtoequationsofthe second order. Theanalyticalconnexion consisted intaking new variables defined bytheequations X=p,T=q, Z=px+qy-z, fromwhich there were derived thereciprocal equations From thesewehave .,.sotnat *. -rtfp_on jrr-Sdx+Rdy**RT-8"' But rdx -f-sdy=dp=dX, 242.]"DUALITY. 377 wetherefore obtain, byequating coefficients, T : -s,R t= 'RT-S" ~RT-S*' "-RT-S" andalso rt s*= Letthese substitutions beappliedtoanyequationoftheform \r+/w?+vt+a-(rt-sa )=0, inwhichX,[i,v,a-arefunctions ofcc,y,z,p,q.Lettheir values afterthetransformations havetakenplacebedenoted byV, //,',v,<r respectively;then theresult ofthesubstitutiongives Ifthen thesolution oftheformerequation beknown, that ofthe latter canbeobtained;and viceversa. Thus inparticular thesolutions ofthetwoequations andTX(.',y)-sty(a,y}+t<f> arederivable fromoneanother. Ex. 1.From thesolution of derive thatof Ex. 2.Integrate theequations (i)px+qy-axy=z; (ii)a(rt-fp)=pq8 ; (iii) }a (z-px-qy)=(pt- qs)xz; (iv)p*r+2pq8+q*t=(xp+yq) (rt-a2 ); (v) Laplace's method forthetransformation ofthelinearequation. 243. Thelinearequation Rr+Ss+Tt+Pp+Qq+Zz=U inwhich jR,S,T,P,Q,Z,Uarefunctions ofxandyonly,canbe reduced tosimplerforms. Theprocessconsists inchangingthe variables. 378 LAPLACE'S TRANSFORMATIONr [243. r f Lettheindependent variables aoandybechangedtoandrj, yetundetermi equation becomesO^ asyetundetermined; then,whenp', </,... denote ^, -,... the das .C j.T.LP j.no~3+'>:>o~5I"-^5~5+-^5~+ Vo' das' dwdy dpoacoy) Letmandnbetheroots ofthequadratic equationink and firstsupposethatthese roots areunequal;then choope fand 77sothat d% 3%=m*r> oac oy 9?7 dv'n- da;dy' which determine ^andrj.Thetermsinvolvingr'and t'now dis- appear ;andthecoefficient ofs',being doesnotvanish since theroots ofthequadraticareunequal.Let theequation bedividedthroughout bythis coefficient; then it takes theform d*z dzTir 244. Intwocases theintegralofthisequation can,without furthertransformation, beobtained. Wemaywrite itintheform 244.]"OFTHELINEAR EQUATION. 379 sothat,ifthecondition besatisfied, theequation becomes %+Mu-V, dzwhere ureplaces^-+Lz. A.generalvalue ofucanbeobtained, andthence ageDeralvalue ofz. Wemaywrite theequationalsointheform 1&+M.} +L(!|+J&)+,(j-LU-*j*}=V, or]V3 J \9f J \ dTjJ' sothat, ifthecondition besatisfied, theequation becomes ^\ where vreplaces^+Mz.From this,through v,ageneralvalue ofzcanbeobtained. 245. Ifhowever neither ofthese conditions between the coefficients inthetransformedequation besatisfied,itcan stillbe transformed bychangingthedependentvariable. Thuswhenwe write wehave or Denoting LM+^N'bjKvre maywrite iarMv-K^K 380 LAPLACE'S TRANSFORMATIONr [245. r r andtherefore Ld{LMLV 9fj.3fJf F) ?~^af+7r^~ 5:+ ai,iBra #5^j which isequivalentto i, r' i-where I/=~ 7f9~' sothatthesame form isreproducedbutwith altered coefficients. Theequationinitsnewform canbeintegrated,iftheanalogous relations between thenew coefficients besatisfied. From the values ofL',M',N'wehave =_ } drj sothat asRisnotzero(byhypothesis),therelation 077 isnotsatisfied. Theother condition, beingthattheequation should besatisfied, iswhenexpressedinterms oftheoriginal coefficients__ L _A+ +a' Ifthisbenot satisfied northecorrespondingrelation derived bytheconsideration oftheotherexpression T,f ,3M ,rLM+-= N dy theprocessoftransformation mayberepeated indefinitely ;and, if atany stepoftheprocesstherequisitecondition should be satisfied, thesolution maythenbefound. ~OFTHELINEAE, EQUATION. 381 \ " .1.Prove thatforanysubstitution oftheform z=pu, uigtobethenewdependentvariable andpisafunction of|and17, LM-N+^andZtf-tf+S^ 9f79 pl>eabsolute invariants andthat therefore such atransformation isineffec- forthepurposeofsolution. v.2.Prove that if Kr=tfr-LrMr-d~randJT=Jfr-LrMr-^dCdl? fuoctions ofthecoefficients after rtransformations) then Heu.ce solve theequation 8+asyp=2yz. (Imschenetsky.) 246. Next, consider thecasewhen theroots ofthequadratic xroeq-ual,sothat The twoequations determiningand17nowcoincide sothat Erom themonlyoneofthesequantitiescanbeobtained;letitbe g,given by 3| 3% a=m^->ox oy a,iid suppose ^andytobethenewindependent variables; then- weinaywrite77=y.^Then inthetransformedequationthecoeffi- cient ofr'iszero, that oft'isT,andthatofs1 is _f_. Butwbeingarepeatedroot of Ekt+Sk+ wehave S POISSON'Sr [246. f r hatthecoefficient ofs'is ch iszero. Hence thetransformedequationondivision jughout byTbecomes Thecasesuitable fortreatment bythismethod isthatinwhich 3zero;theequation maythenbelooked uponasanordinary Lation iny,thevariable xbeingconsidered constant; the itraryconstants ofintegrationshould bereplaced byarbitrary ctions ofx. Poisson's Method. 247. Poisson hasshewn how todeduce aparticular integral mypartialdifferentialequationwhich isoftheform P=(rt-s^Q, erePisafunction ofp,q,r,8and thomogeneouswithrespect thelastthreequantities, andQisanyfunction ofthevariables y,zandthedifferential coefficients ofz,which remains finite enrt s2=0. Heassumesq=$(p), Itherefore s=rfi(p)and t= s<f>'(p}=r{<'(p)}a . These values make rt s*= ireduce thedifferentialequationto P-0. NowPbeing homogeneouswithrespecttor,sand ttthere will, Lentheforegoingvalues aresubstituted, occur acommon factor roughout, being somepowerofr;thismayberejected andthe naining equationwillinvolveonly p, <f>(p)and<'(p)whichwhen jegratedwilldetermine thevalue of </>(p)andsowilllead toan ;egraloftheoriginal equation. Thisintegral, beingoftheform ?=(Pi D.always befurtherintegrated, 247.] METHOD. 383 Itmay benoticed that Poisson'sprocessisequivalentto obtainingthedevelopablesurfaces which areincluded under the givendifferentialequation,for isthedifferentialequationofdevelopablesurfaces. Ex. 1.Solve ra-Z3=rt-a*. Proceedingasabove wefind 1-{0}*=0, sothatretaining onlytherealvalues *'(P)=1, whenceq=<(p)=ap, where aisanarbitrary constant. Thecomplete integralofthisconsidered as apartialdifferentialequationofthefirstorder is where Xandvarearbitrary constants;thegeneral integralis where isanarbitrary function. Ex. 2.Solve (i). (ii) LinearEquations with constantcoefficients. 248.Wenowproceedtoconsiderequations which arelinear notmerelywithregardtothedifferential.coefficients ofhighest order butalsowithregardtothedependentvariable and allits differentialcoefficients, andinwhich thevarious terms aremulti- pliedbyconstantsonly. Such anequationis 3>/9 3\ TT (a^HF where$isarationalintegral algebraical function allthe coeffi- cients ofwhich areconstant; Vmaybeanyfunction ofthe independent variables. 384 LINEAE EQUATIONS'[248. * Asinthecase ofordinarydifferentialequationsthecomplete integralconsists ofthesum oftwoparts: first,themostgeneral integralof second, anyparticularsolution of These will beobtainedseparately.Forconvenience, let O O -and5-berespectivelydenoted byDand D'. 249. Thesimplestcase ofthegeneral equationisthat in whichonlydifferential coefficients ofthewthorder' occur, sothat it maybewritten (Dn+AJT*jy+Ajy~*D11+......+AnD'n )z=V. LetOj,aa,......,anbethenroots of thentheequation maybetransformed into (D-^D") (D-2aD')......(D-aj)1 }z=V. Tofindthecomplementaryfunction wewriteV= ;then a solution of willbeaterm inthecomplementaryfunction;andasthere are nsuch factors there willbensuch terms. Now thesolution of (D-\)z=0, where \isindependentof#,isgiven by beingalsoindependent ofx.Thequantity Cmay therefore, in thesolution of 249.] WITH CONSTANT COEFFICIENTS. 385 bemade anarbitraryfunction ofy,andwethenhave =<j>(y+CM;). There isonesuch solution forevery value ofa;andthesum ofthese different solutions isalsoasolution, sothat thecom- plementaryfunction is wherefa,fa,......,faareallarbitrary functions. Inthecase,however, inwhich tworoots aareequalthisvalue ceases tobegeneral,asthesum oftwoarbitrary functions ofthe sameargumentismerelyanarbitraryfunction ofthatargument; thecorresponding terms, arethenobtained asfollows. Thesolution of s z=e whereAandBareindependentofx;hence theintegralof is z=e t =$(y+cuK)+x-^r(y+ow?), where both(f>andi/rarearbitrary ;thesum ofthese twoterms replacesthesum ofthetwoterms, which hadcoalesced intoone, andthegeneralcharacter ofthesolution isrestored.Similarly, when anynumber oftheroots aareequal,thecorresponding terms ofthecomplementary function, which coalesce into one, arereplaced byaseries ofterms derived inthesamemanner as theabove. 260.Toobtain theparticular integral wemay representit symbolically by 1 TT z= V.'D_ F. 25 386 LINEAR EQUATIONS" [250. Toevaluate thisweresolve thesecond symbolicalfraction into thesum ofnsymbolical partial fractions, intothedenominator of each ofwhich onlyoneofthequantities D/D1 a.enters ;thus, if 1r=nN_~r_ wehave 1r=n ATs-i27^-F JJraiU ~D'-*' =*_/!*N'V, Nrbeingaconstant anddepending onlyupontheconstants a. Let V=ty(a>,y); then since (D-oD')'1=e*** wehave r =j hence theparticular integraloftheequationis y+a, (-0}]. r-l This isthevalue inthemostgeneralcasepossible ;inparticular cases theactual evaluation becomes much moreeasy. Thus, ifV beafunction ofasonly,wemayconsider[4>(D,DO}"1^expanded inaaeries ofascending powersofD'andtheneverytermmaybe neglected (sofarastheparticular integralisconcerned) except thatwhich does notcontain D'.Corresponding simplifications arise inotherexamples. 150.JWITH C1HSKTANT LI * *. t'Jt* ryf iSroV.\. 1,i^l.J FIT tlir <'i.iMl'li'iiu'ntflryKuiirtinit woIwvo ('"Mf'+"' \*-"Vr jr/V-r i^/ ilnlthi-n-f'Tf *** .nut^li-uiitorlulwy. K>-r tlirI'nrtii-uUr ItitagnJ wehave t "*//< //' -M r1 ~3V llrinr IIIPt'ii|ii'li-{ /.*! J(,UllUlll II<<hll|ll|l llflilt*<N}IUlLtll|l tluit. .(7 iM*ri, .A Siilvft Ihfn|uat|<>lm (v) fri) 388 LINEAR EQUATIONSp[250. r r Ex. 4.Tosolve 33743%3M _ 5-5+5-5+a-q-3 3a?say33s3 FortheComplementary Function wehave 33\/3 3 abeing aoube rootofunity. Thesolution of hence theComplementaryFunction is where1}$s,8arearbitraryfunctions. Thepart oftheParticular Integral correspondingtoaflis 1 1.afl andBOfortheother terms;thefullvalue is 4.5.6+~!T' TheComplete Integralisthesum oftheComplementary Function and theParticular Integral Ex. 5.Solve 3^M 3^it3^ 3"w d ~ 251.Passingnow tothegeneral equation, wemust findthe solution of I-' }*=<>>patMj 251.1"WITH CONSTANT COEFFICIENTS. 389 where <I>isoftheform " i 1Weassume asatrial solution where handkareconstants yettohedetermined;forthisvalue, fai j^z7*-=hzand^-=fez; da; oy andtherefore wehave <S>(h,K)z=0, which willbesatisfied, ifhandAbedetermined soastosatisfy This obviouslymakes oneoftheconstants todependonthe other;lettheequationbesolved todetermine k,sothatweshall have results oftheform ninnumber. Takingone ofthem, ask=-01(h),wehave the solution intheform, forallvalues ofAand h.Nowthesum ofanynumber ofsolutions isalsoasolution, sothatanother isgiven by. where 2impliessummation for allvalues ofh;andA,an arbitrary constant, maybelooked uponasanarbitraryfunction of hwhichmayvaryfromterm toterm oftheseries. Similarlyanother value ofk,such asa(K),willlead toanother solution whichmayberepresented by 390 LINEAR EQUATIONS p/ [251. and, aseach value ofkwilllead toacorresponding aeries, the generalsolution mayherepresentedasthesum ofnseries inthe form thesummation ineach series extendingtotermsarisingfrom all possible values oftheconstants h.The factthatthecoefficient "belongingtoanytermmaybeconsidered asanarbitraryfunction oftheconstant which occurs inthatterm shews thateach series mayberegardedashavinginitsexpressiononegeneral arbitrary function;andthus intheComplementaryFunction weshould be ledtoexpect narbitraryfunctions. 252. Thisgeneralresult intheform ofthesum ofnseries eachcontaining arbitraryelements mayappearbobeofslight value. Sometimes, however, bytheform ofthe differential equation, asimplificationisintroduced such asthat indicated inthenextparagraph ;sometimes byconditions imposedonthe dependentvariable other than thesatisfaction ofthedifferential equationthenumber ofterms oftheseries islimited tothose which containparticularvalues oftheparametricconstant. Forexample,whenever asolution oftheequationwhich determines A;isoftheform where aand{3aredeterminate constants, thecorrespondingseries maybeexpressedinafinite form. For itis thatis,itis(saveastothefactor outside S)thesum ofany number ofarbitrary powersofemJfttyeachmultiplied byanarbitrary constant;such asum isanarbitraryfunction ofF+ay or,what is theequivalent,anarbitraryfunction ofac+ayandtheseriesmay therefore bereplaced by where<f>isarbitrary. Correspondingtotheconditions which in anyparticular case limit thenumber ofterms included inthe series, there willbeanalogousconditions which determine the form ofthearbitraryfunction. 252.] WITH CONSTANT COEFFICIENTS. 391 t Ex. Prove that,iftheroot occurr+l times, thecorresponding partoftheComplementary Function is where<,<15......,(rareallarbitrary. 253.Toobtain theParticularIntegral wemay represent itby g_1rr. ' ' theevaluation ofthisexpressionwilldepend upontheform ofV. Thus if V= etuc+bv , weshould have asthevalue ofzrequired.IfVwerearationalintegral algebraical function ofCDandy,then itwould bepossibletoevaluate theex- pression byexpandingtheinverseoperatorinaseries ofascending powersofbothDand D',ifpermissible,orofoneofthem. The methodsappliedtotheparticularforms considered in46inthe case ofordinarydifferentialequationswill indicate the corre- sponding methods tobeadoptedforthevaryingforms ofV. Ex. 1.Solve d*e _9s _9z -5-5-3 5-+35- djroxoy First,fortheComplementaryFunction wemust solve Let besubstituted;then (A sothat k=handi=3-h aretherelations between h.andLHence where<j>and^areboth arbitrary. 392 LINEAR EQUATIONS' [253. r f ThepartoftheParticular Integral correspondingtoeP+sis Df(ff+lY The result indicates thataterm oftheform ea+3 fwill arise intheCom- plementary"Function;that this issoisobvious from theidentity ThepartoftheParticularIntegral correspondingtoayis theexpansionsineach casebeingtaken nofurther than isnecessaryto furnish non-evanescent terms. Itmight happen that,byadifferent method ofprocedure such asexpandinginpowers'ofTyaparticular integralof apparentlydifferent formwould beobtained;itwould however befound that thetwocouldbetransformed intoeach otherbymeans oftheComplementary Function. Thegeneral integral is,asusual, thesum oftheforegoingthreeparts. 254.Anyequationsuch thatthecoefficient ofadifferential coefficient ofanyorder isaconstant multipleofthevariables of thesamedegree maybereduced toanequationoftheforegoing form. Suchanequationwillbeoftheform 254]WITH CONSTANT COEFFICIENTS. 393 Wemayeither changetheindependentvariables touandvwhere =e";orwemay represent0ty *andVbv*' andthenwehave 0-|fl=*(*-!)(*-- 2)...(Sr- oar OSS0^ Ineither case theequationisreduced totheform alreadycon- sidered Ex. 1.Tosolve Wehave,onassuming u=loga;andv=logy, /3J- ts+ Theintegral ofthis is where/andFarearbitrary. J&CT. 2.Solve p.3.Solve theequations r.4Solve 394 MISCELLANEOUSr [254. r f Ex. 5.Solve ,.., /B3^3a*\., ,.ft /3s 3s\ (u)ro[^ +^)-(ma+8)3-3-+wm[ji- 5m*--)VMT oy*J^ 'fooy \da;oy/ =cos 32 Ex.G. Solvef(-ar)z=ff n, 7\ ftrs where ordenotes theoperator^5+a;s+...+xm*,/isarational inte- 1 2 Hi gralalgebraical function ofm,andHnisahomogeneousfunction ofndimen- sions ofthequantitiesa;lt#3,...,xm. Miscellaneous Methods, if 255. There areseveralpartialdifferentialequations which areoffrequentoccurrence inphysical investigations; solutions ofthese havefrequentlybeen obtained bymethods, theappli- cation ofmost ofwhich toequationsother than those inconnection with whichthey originatedisverylimited. Thetwo chief methods areintegration bymeans ofdefiniteintegrals and inte- grationinseries; but aseachmethod isofspecial application only,andasthevariations which ariseowe theirorigintothe conditions imposed uponthefunction whose value issought and nottoanyvarietyinthedifferentialequationstowhich itcanbe applied,itisnotpossibletogivehere afulldiscussion. The dis- cussion here willbelimited toafewexamples;forfuller investi- gationsrecourse must*be hadtothetreatises onthose branches of mathematicalphysicsinwhich thedifferentialequationsoccur. 256. Consider firstanequation which canbeintegrated by bothmethods. Such anequationis du o9!M 256.]'METHODS. 395 ,arises. _in.investigations connectedwjth^ theconduction _of Itisnotwithout interest toindicate thedifferent methods whichmaybeappliedtoobtain asolution. Bythemethod of249wemaywrite u=e where <(oc)isarbitrary;expandingthedifferentialoperator we obtain u=<PW+^^+-2T^-^T^-- sothat thesolution contains onearbitraryfunction. Wemay proceed otherwise thus :thesolution of d*u . s u=A+e- t whereAandBareindependentofx;sothatwemayexpressthe solution of 9V_1du do?~afdi intheform wheretyand^arearbitraryfunctions. Inorder tofreetheresult fromsymbolical operations,which wouldrequire interpretationif theyremained, wechangethearbitraryfunctions to/andF,where -%(*)}; then since^andvarearbitrarybothfandFwillbearbitrary, ,j 1.1 whateverinterpretation beassignedtof-^J.When thesym- bolicaloperatorsinthe firstform ofsolution involving tyand% areexpandedandtheterms ofthesame order indifferentiation aregathered together,thesolution becomes 396 MISCELLANEOUS* [25 t, a?Ai tfdV andthiscontains twoarbitraryfunctions. 257. Itmayatjfiratsightseemparadoxicalthattwoperfectl; generalsolutions ofthesame differentialequationcanbeobtainec ofapparentlysodifferent acharacter. Thedifficultywill dis appearifitbenoticed that theequationisonlyofthe firstorde intwhile itisofthesecond order inx;theformer solutioi containsonlyasingle arbitraryfunction ofac,which isallthat cai beexpectedinthecase ofanequationofthe firstorder;th( second solution contains twoarbitraryfunctions oft,which isth( number ofarbitraryfunctions tobeexpectedinthecase ofar equationofthesecond order. Ifweassume that allthearbitraryfunctions canbeexpanded inpositive integral powersoftheirarguments, weareable tc transform oneofthese solutions intotheother. Forlet where thecoefficients Anarearbitrary,and letthisvalue besub- stituted inthe first solution. Then thetermindependentof is which isaseries witharbitrarycoefficients andsomaybedenoted where /isarbitrary;thecoefficient of(-)=-is \Cb/ft 1 21 -f\etc partofthesolutiondepending upontheevenpowersofasisthat is,-f\and sofortheother evenpowersofx.Thus the etc i 257.]* METHODS. 397 r> o Similarly collecting thetennadepending upontheoddpowers ofxandwriting (whichisanotherarbitrary function) weshould obtain thesecond partofthesecond solution. Itthusappearsthatthetwoalge- braical expressionsareequivalent, independentlyofthefactthat theyareboth solutions ofthedifferentialequation. SolutionbyDefinite Integrals. 258.Now letthemethod of251beapplied. Wesubstitute u=eaa+llt ; thenecessary relation between theconstants aand/3is sothat u=Ae*+atatt , forallvalues ofAanda,would beasolution. Instead ofawrite aisothat solutions aregiven by andtherefore by whereXisanyconstant andAandBarearbitraryfunctions ofX. Thesemaybereplaced by >-X andBfe~a whereA'andE'arearbitrary functions ofX.Further thesumof anynumber ofsolutions JBalsoasolution. Consider thatobtained bysumming anynumber ofterms oftheform ofthe first forall values ofXandaandassumingthatwhile A'isanarbitraryfunction ofXtheformofthearbitraryfunction isthesame fordifferent values ofX.(The correspondingterms which would arisefrom the- secondmaybedeemed included inthissince sofarasthevariable partisconcerned weneedonlytochange XintoX=-toobtain thefirst.) Letthen A'= *//(X)cZX, 398 SOLUTION BT[25 * * -andsuppose summation totakeplaceforallvalues ofXbetwee ooand+oo;thecorrespondingsolution is f* e'***cosa(0-X)i/r(\)d\. J OB Thisagainmaybemultiplied byanyfunction ofaand th summation taken forallvalues ofa;asitstands thefunction anevenoneofa,andsoifthefactor betaken asda.itwill suffic totake and ooasthelimits ofa;andthuswemaytake asth solution rrda I J-t Thesolution inthisform isspeciallysuitable forthecaseinwhich uis 1 satisfy some condition,forinstance that -/(*)' when iiszero;thuswearetohave /GO rat /(ic)=eZa Ioosa(x-\)-^(\}d\.Jo J- But,byFourier's theorem, thevalue oftheright-handside ismfr(#)soth \ffisdetermined;andthus ir r ?rJ J-oo r(Kiemann.) Ex. Obtain a*solution oftheequation, =a2^, 'which issuch that u=f(x)and -~.=F(a?), when t=0. Theresult is (Riemann.) 259.Wemayagainsolve theequation byamethod, du> originallytoLaplace andextended byPoisson. Wehavebyaknown theorem r.-**-**, J-oo 259.] DEFINITE INTEGRALS. 399 e or,writing u Iforuwhere Iisindependentofu, When Iisanydifferentialoperationtobeperformedthis relation indicates thatthesymbolical operationepcanbeexpressed providede2^canbeexpressed. Thismethod maybeappliedtotheequation du_ ad*u di~afa>] forwehave where/(a?)isanarbitraryfunction independentof t.The fore- goingformula inequivalent operators maybeappliedifIbe j replaced bya-j-;andthuswehave CWD M=7r"iPJ-00 Another formmaybegiventothisresult bysubstitutingA,for x+2wat*. Thenwbecomes Now/(X)isanarbitrary function;ifwechoose toassume its value tobezeroeverywhere exceptwhen \=randthenwnte /(X)d\=H,wehave JSii?. 1.Provethat,ifusatisfytheoonditiona (i)u=f(x) when i=0, (ii)u= DEFINITE-INTEGRAL SOLUTIONS. >rL then itsvalue is"' Ex. 2.Obtain asolution oftheequation intheform Ex. 3.Verify that 1 satisfies thedifferentialequation ' and issuch thatwhent=Qthenu=F(x, y,g)and^=/(#, y,*). ii ot 4.Obtain thevalue oftheintegral ff taken overthesurface ofasphere whose centre istheorigin andradius theform R4fr- where a= Hence shew that themean value over thesurface ofanysphere function, which satisfies theequation andis,forallpoints within thesphere, expressible byaconvergent ser equal tothevalue ofthefunction atthecentre ofthesphere. Further information onthispart ofthesubject and,inparticular, o applications inphysical investigations, willbefound inEiemann's Pa: undderenAnwendimg aufphysi&alische Fragm. SOLUTION INSERIES. 4Q1 SolutionmSeries. 260. Consider nowacaseofintegration bymeans ofseries. Themostimportant equationtowhich thismethod isapplied istheequation 9V9V9V_ whichcontinually occurs inphysical investigations ;tosolve itbythemethod under consideration itisconvenient tochangethe independent variables from a,y,ztor,6,<given bytherelations a>=rsin6cos0,y=rsin sin<,z=rcos9, which willineffect bechangingfrom theCartesian tothepodr coordinates ofapoint. Theequationisnow _9a (ru),19/..du\ ,19V n and, ifanotherchangebemade bywriting pinstead ofcos d,the resulting form is rtf(ru) _3_ f._,du\19V= 261. First, letasolution bedesired which isto-be afunction ofronly, that is,of(#*+y*+^l )i ,sothat itwillbeaspecially symmetrical solution;theequation then reduces to n andtherefore u=A -\. r Inasimilar-wayasolution which would beafunction of&alone, andonewhich would beafunction of alone,maybededuced; buttheyarenotsouseful asthatjustobtained. 262. Next, supposethat solutions which arenotfunctions of ralonemaybeexpandedinaseries ofintegral powersofr;and inuletthere beaterm F.'26 402 SOLUTION-[262 n where unisindependentofrbutmaybeafunction of6and <the value ofwhich isstilltobedetermined. Then,when thevalue oi uissubstituted, theterm ontheleft-hand side ofthe differential equation correspondingtothisparticularterm ofuis andthesum ofallthese terms istobezero forallvalues ofthe independentvariables. Theforegoingistheonlyterm which involves then^powerofr;ittherefore follows that, inorder to have theequation satisfied, itscoefficient must vanish. Hence un isdetermined by andtherefore rnunisasolution oftheoriginaldifferentialequation. The coefficients ofthetermsinvolvingthedifferential coefficients ofundonotdepend uponn;andthecoefficient ofunisunaltered iffornthere besubstituted(rc+1);hence r~(n+i)unisanother solution oftheoriginal equation.These two solutionsjust ob- tained maybecombined intoonesoastogive .. .i/n' asasolution, AnandBnbeing arbitraryconstants;andthusthe generalvalue ofuis =J(4,^+pHiJw providedunbedetermined bytheequation 263.Nowthegeneralsolution ofthisequation wouldgiveun asafunction of6and<;consider thecase inwhich unisa function of6only.Itisthendetermined by 263.] INSERIES. 403 > theindependent particular integralsofwhich are(90,91)Pn(p) Qn(/*) 5thecorrespondingterms inuare Inmostphysical investigationsthetermdependent uponQn(//,)is rejected ;andthenthegeneralvalue ofu,expressedasafunction ofrand0,that isofzand(0a+y*~?,is u= inwhich theA'Band B'earearbitraryconstants. Itwillbe noticed that thesolutionformerly obtained, viz., 13theparticular caseobtained bymakingallthesearbitrarycon- stants zeroexceptAandBandrememberingthatPO(/A)isa constant. 264. Consider nowthegeneralcaseinwhich unisafunction of6and <;itmaybeexpandedinaseries oftrigonometrical functions ofmultiplesof(bthecoefficients ofwhich arefunctions ofp.Anyterm oftheseries forunmaybedenoted by where visafunction ofponly; and,justasinthecase ofthe separateterms inuconsidered asinvolvingdifferentpowersofr when eachsuchtermwasasolution oftheequation,thiswillbea solution oftheequation givingun.Substituting anddividingout bycos cr<wefindthat vn(<r]isdetermined bytheequation Thisequationwould alsohave been obtained bythesubstitution intheunequationof andtherefore thesolution oftheequationinunis '" {Evsino-c/>+F,,cosa-0}vn(<r] <r=l 262 404 SOLUTION"[264 n r thevalue a-=notbeinghere included, since itgivesterm; independentof <which havealreadybeen found. Now,byEx.12,Chap, v.,p.180,thesolution oftheequatioi givingv^is where ynisasolution oftheequation when a-iszeroand thui maybeeitherPnorQn.Hence thecorrespondingterm inunis ovio-dP_sno-<>+ cos o-< (E'asino-0+-?"cosa a/A Theterminvolving Qnisusually rejectedinphysicalinvesti gations ;thesuitable value ofunthen is T(1- /Lt8 )4'(Evsin<r0+J^coso-^)^= , itbeing obviouslyuseless toinclude values ofa-higherthan n. Thesum ofanynumber ofsolutions oftheoriginal equationi; asolution;andtherefore themostgeneralvalue ofuexpressedii aseries is n=to +2 n=l +2X(1-A'S + sin~ Wehave omitted from theforegoing general value(1)tht terms which would arise from thepartofuindependent ofranc <j>,which caneasilybeprovedtobe (2)thetermdependent upon$alone whichobviouslyisM$,anc (3)thetermsusually rejectedasunsuitable inphysical investiga- tions. 264.]-INSERIES. 405 i Any furtherinvestigations onthesolution oftheequation are connected either with otherequivalentforms ofsolution orwith theparticular solutions obtained byadetermination ofthecon- stants inaccordance withimposedconditions. Forthese recourse should behadtotheauthorities ontheseveralsubjectsinapplied mathematics inwhich thisequationarises;inparticular, those quoted onp.159willbefound ofgreatvalue. Ex. 1.Solve theequation inseries, bytransforming topolarcoordinates. Ex. 2.Prove thattheequation <Pu_ n/B2^3^W~\dafl+ 3y*+ hasasolution oftheform where yw-*-raT2.4.a2 2.4.6.28' 1.2. 3...Zn* 2.4.6...27i.**' Obtain amoregeneralsolution which isnotindependentofthespherical coordinate <. (Stokes.) Ex. 3.Shew thatthegeneralsolution oftheequation a2U-d+= or,bytransformation toplane polar coordinates,itsequivalent &u 1dul&u\&u canbeexpressedinterms ofBessel's functions asthesumoftwoterms ofthe form M=cosofeT [{AJn(*r)+5F M(*r)}cosn6+{A'J n(4r)+5T n(*r)}sin nfl. 71=0 406 AMPERE'S[265. Ampere's Methodofsolvingtheequation -f=V. 266. There isanother method ofproceedingfrom the dif- ferentialequationtotheintermediary integralinthecase ofthe general equation thefactor 2beinginserted forconvenience. Letanewindependentvariable a,asyetindeterminate, be introduced and letxandabeconsidered astheindependent variables sothatyisafunction ofxand a.;thenwehave dz_ dydz_dy dx~P+qdx' ~fa~qd~z' dp dy dp dy-=r+s1f, -/-=s^-,dx da da da. d(i-s tdyd(i-tdy ~T~~~~t>T fr~T~ I "l~~~"~T~dw dx da. da. j j Biere -=-and -=-areused toindicate partial differentiation das dy.c withregardtothenewindependentvariables xand a.From these equations wehave dp dyT~~S da; dx fyt=^-8 das dx' aai cuedec inallofwhich sistobereplaced by dp dy^ do.' do.' When these values aresubstituted intheoriginal equationit takes theform 408 AMPERE'S [266. t form, butnotindependent. Multiplying (ii)byZ7andsubstituting from(i)forU-~.wehave whicheasily reduces to U^-(S+G*)^+T=0................(iii).ox ox Wemaythus consider either(i)and(ii)or(i)and(iii)asthe equations whichreplacethetwoP==Q.Takingthen(i)and (iii)wemayrewrite them intheform Udq+Rdy-(S 0*)da= andwehave also dzpdcc qdy= inwhich itwillbenoticed thatdadoesnotoccur andtherefore a istobeconsidered aconstant intheintegrations. 267. Thesuccess ofthemethoddepends upon thepossibility ofobtainingafunctionWofCD,y,z,pandqwhich shallbesuch that, invirtue oftherelations between thedifferential elements expfeasedbytheequations (iv),itstotal differential shallbezero. Ifthisbepossible, wethenhave JTI7,J , J, , Jn dW=-=- das+-z-dy+-^dz -\--^-dp-{-5da=: deeay*ozoprdq* when thevalues ofdz,dp,dqasgiven by(iv)aresubstituted in this,itbecomes anequation involving onlythetwo differential elements da;anddy,which areindependentandthecoefficients of which must therefore beseparatelyzeroinorder thattheequation maybesatisfied Thuswehave Either ofthesemaybereplaced by 265.]METHOD. 407 * wherePandQaregiven by p==Rdpdy+Tdq+ufydq_v<fy dxdx das decdas das' 2S< j>das Asyetaisarbitrary;letitbechosen sothatPvanishes; then itfollows from thedifferentialequationthatQalsovanishes and thuswehaveP=0,Q=0. 266. Theseequationscanbereplaced bysimplercombinations equivalenttothem. From the firstwehave - dx\ das dec! dx dx' when thisvalue of-^issubstituted inthesecond equation the dot latter becomes afteraslightreduction-- dx dx) \dx whichgives R^+U^=SQl> ..................... (i)' where G=&-RT-UV. Thecorrespondingvalue of-pisgivenby ^(Sffi)=F |_r|, or,what isthesamething, andtherefore R^+(8+fl*)^|=V. (^)-* Theseequations (i)and(ii)mayreplacethetwo firstobtained; itwillbenoticed thattheyareanalogoustothose in234.We mayalsocombine(i)and(ii)soastoobtain anequationinanother 26*7.]METHOD. 400 which results from theelimination of-=between thetwo,and dq division byU. This lastequationhasbeenobtained ontheapparent supposition thatUiszero. ButinthecasewhenUiszero itiseasytoderive itfromtheequations _ decdx dec da; dz=pdtc+qdy, bysubstitutingfordas,dy,dzinterms ofdpanddqintheequation. dW=Q andequatingtozerothecoefficient ofdp.Theequation canthusbeused injihecasewhenUiszero; thetwoformer equationsareinthat caseequivalenttoonly one,which would be combined withthenewequation. ThefunctionWmust thereforesatisfytwosimultaneouspartial differential equationsofthe firstorder;themethod ofobtaining such asolution common tothetwo,when itisknown toexist, is indicated in226andwemaytherefore nowconsiderWaknown function. 268.Asolution ofthegiven differential equationisfurnished tyW=constant. Forwethenhave dWdWdWdW . Q+-~-p+-~r+-3-s=0, das dz op oq 9FdWdW. dWdW andthese, onthesubstitution inthem ofthevalues of-~and -= 'ooo oy from theforegoing equationswhich determine TF,becomerespec- tively 410 AMPERE'S METHOD. [268. n r dWdW.Theelimination oftheratio of-~to-^between these gives dp aq (T+Ur)(R+Ut)=(S-Us)*-G, which, invirtue ofthevalue of0-,reduces to Rr+2Ss+Tt+U(rt-s*)=V, thatis,totheoriginal equation.Thepropositiontherefore follows. 269. Inorder toobtain themostgeneral intermediaryinte- gral,wemust findanexpressionwhich contains anarbitrary function. Supposenowthat itispossibletoderive twoparticular solutions wlandwaoftheequationswhich determine W,and which are,owingtothedoublesign, reallytwo sets; then the equationswillbesatisfied bywriting W=$(wlfwj=0. Since theequationsinWarelinear this isobviouslyasolution. Alsotheparticularsolutions are w^=constant; butintheintegrationswehad toconsider aasaconstant, and therefore wemaywrite ^=/, (a), where/j(a)isanarbitraryfunction.Similarly weshould have where/8(a)isanarbitraryfunction. Nowaissome function ofas andy,thevalue ofwhich isunknown; whenwesubstitute in eitherequationthevalue ofaderived fromtheother,weobtain a result oftheform indicated. 270. Itmayhappenthatmore thanonegeneral intermediary integralcanbeobtained. Inanycaseweproceedasbefore from thesingle intermediary integral (byCharpit's method)orfrom the combination ofthetwointermediary integrals (asin236)tothe general integraloftheequation; and thisintegralwillusually involve either twoarbitraryfunctions orthreearbitraryconstants. Thishowever isnotthemostgeneral integral possible. For ifwe hadanoriginal integral equationoftheform <f>(z,as, y,a t,as,aa,o4,OB)=0, andobtained thence fiveotherequations givingthevalues of p,q,r,s,twecould between thesrsresulting equationseliminate 270.]GENERALISATION OFINTEGRALS. 411 1 thefiveconstants aandhaveadifferentialequationofthesecond order;andaccordingtotheform of <j>thedegreeofthisequation wouldvary. Converselyinanycasewemightinthat integral, which ismostgeneralsofarasthenumber ofarbitraryconstants which enter isconcerned, expect more than three. But$= will notnecessarilybethemostgeneral integral;theonlyinference to bemade isthattheequation containingthreearbitraryconstants isnotthemostgeneral integral.Itcanbereplaced however by onewhich ismoregeneral;themethod ofobtaining this,due to Irnschenetsky,issimilar tothatemployed byLagrangeforpartial differential equationsofthe firstorder viz.,variation ofthecon- stants. 271. Lettheintegralobtained bytheforegoing method be represented by z=f(ec, y,a,b,c) ; toobtain thegeneral integralweshallsupposectobechanged intoafunction ofaand bthevalue ofwhichis,asyet,undeter- mined andthen consider aand btobefunctions ofa;andysuch thatpandqpreservethesameformsaswhen a}b,careallcon- stants. Denoting + and da dcda db dodb . . ,df Adf, respectively by-j-and^,wehave dz_= df_da_ dfdb dx~" da,dasdbda' dz_ dfdadfdb dy=q+~dady+ ~dbdy] dz dzand therefore, since=-=pand5=q,weJhaveox^o =idadxdbdss dfda+dfdb=Qdadydbdy' which willbesatisfied ifwewrite -"-&da do 41-2 GENERALISATION OF[27l. Thesecond differential coefficients are 9*z=dpdadpdb_, dx*r+dada;+dbfa~r+' d'-zdpdadpdb_dqdadgdb_ =+ + -8+ + -8+k &*_.,dgdadqdb55=C-t-j=--r-jT-=5+6. o^T aa3yd6ty dfBut since-^-isidenticallyzerowhen wesuppose aand replaced bytheir values interms ofasandy,wehave 9/d/\ ff/3a J/db do:(da)^da>da"*"dadb das~ ' and l da\dxj da' sothat ^ ,dffdad*fdbdb^^'"' dq dadbdydb*dy Theseequations satisfythecondition jc_d#da dpdb_dgda dqdb dady+ dbdy~'dadiD+dbfa' andfromthem there canbeobtained theexpressions -- 9\dbj dadbdadb^ db9\da)' Sfl^-ffi-ii+^db) dadbdadb db*(da ^f^dq_^f_(^dqdqdp\da?dbdbdadb(dadb^dadb) where-- da*db*' 271.] INTEGRALS. 413 \ *Butwiththemodified forms ofa,b,c z=f(tE>y>a.&,c) isstilltobeasolution oftheequation 3Vn(<?zay /vzYI5-5+L/1-5=- L,-\YV\ * * ' thecoefficients ofthesecond differential coefficients areunaltered inform, sincewehave retained theforms ofthe first differen- tial coefficients, andtherefore R,S,T,U,Vremain unmodified. r\9^vgA Substituting now inthisequation thevalues of,- - dordxdy dy* andrememberingthat thedifferentialequationissatisfied when h,k,Iarezero,wefindthat ittakes theform Ur)l+U(lh-J<?)=V, where thequantities r,s,twhichexplicitly occur andthequantities p,q,zwhichimplicitlyoccur aretobereplaced bytheirrespective values derived from theintegral z=f(, y,a,6,c) inwhich a,I,Gareconsidered constants. Wemustnow substi- tutetheexpressions found forh,k,I;andthentheequation,after some reductions, willbefound tobeoftheform where inallthese coefficients thequantities z,p,q,r,ts,taretobe replaced bytheir values interms ofxandyasderived from the given integral equation. 414 GENERALISATION OF[271- <> r This differentialequationislinear inthesecond differential coefficients of/withregardtoaand b;itis,moreover, the equationwhich istodetermine thevalue ofcasafunction ofa- and b.Now da, 'da dcda' sothat-js=5j+^-5--~r^ uioact od ottocoa oc\oat and also fortheother coefficients;when these aresubstituted for I.=4r, -W-utheresulting equationislinear inthesecond dor'dado do* differential coefficients ofcwithregardtoaand b,andthe quantities multiplyingthese arefunctions ofa,y,a,b,c,- ,^r. Butwealsohave da, db' from which thevalues of scandycanbefound asfunctions of a,b,c,x-,*r',andthesewhen substituted willmake theequation Idado onewhich involvesonlythequantities a,b,candthedifferential coefficients ofc.Thisequationwillthenbeoftheform 3ac^3'c where A,B,G,Farefunctions ofa,6,c,5- ,oi. octoo Now itmaynotbepossibletointegrate directlytheoriginal differentialequation,while itmaybepossibletoobtain, almostby inspection,aparticularsolution which involves threearbitrary constants;oritmaybepossibletoderive suchanintegralwhen notobtainablemerely byinspection.Ineither easesuchparticular integralcanbegeneralised providedthesolution oftheequation tobesatisfied byccanbeobtained;and ifthissolution berepre- sentedby 0(a,&,c)=0, INTEGRALS. 415 _then thenewintegral oftheoriginal equationisobtained from =e(a,6,c) =9/a0_9 3a3c 3cda _ db3c 3c36J byeliminating a3&,cbetween them. .Ek. 1.Integrate theequation HereE=l,S=q-x, T=(^-x}\ 17=0,V=q- thusG=0, andtheequa- tionsdetermining TFareonlyasingle pair, viz. 3FT .3PF Wedenotethese, asin226,by Q=I'z=Z+(q-a;) Y+(p+q*-qx)Z+qP. Asacondition that theseequations maybeintegrated simultaneously wemusthave Hence wewrite Q=F,=-qZ~Y; then (^,^ 3)=0; (FvFj=Z, andsowetake Q=Ft=Z, andthen 0=^,FJ=...=(F 3,FJ. Hence Y=Q=Z; X+qP=Q; Q-(q-x)P=Q; substitutingin Q=Pdp+Qdq+Zda;+Zdz+ Tdy weobtain 0=P(dp-qdx+qdq-xdq). andtherefore wemaywrite astheintermediary integral Toobtain thecomplete integralofthisweapply Charpifsmethod;we must obtain anintegralof _dp_dq ~ -q~ 416 GENERALISATION OFINTEGRALS. This isgiven byj=)3 ;andtherefore These values, substituted in dz=pdx +qdy, lead totheintegral 2=&y+bfa (x~0)~vx~ > which contains three arbitraryconstants. Toobtain themodifiedintegral (271)wewrite this e=f= -ax+0y+$fia: (x-&)-c, consideringcasafunction ofaand)8.Thenwehave ndf9o 0=-f=-x--^;do. oa Hence^=0;^=1 ;^=0;Fj=0 ;andtheequation in/is or,onsubstitution interms ofo, df&~' 9ac3c orfinally 5^=5-. Off Od Butanintegralofthisis,by 259, f(O J-a, andtherefore anintegraloftheoriginal equationisgivenbytheelimination ofaand)3between *4j3# (x 19)-Ie~j J-00 Thesecond oftheseequations may,when thedefinite integralisintegrated byparts, bereplaced by =#-P J EXAMPLES. 417 Ex. 2.Integrate '--*' (ii)aft--tePqs+4/pt+Zpx3= ; (iii) (iv) (v) (vi) (vii) (viii) (Ampere andImschenetsky.) Afuller discussion iscontained inthevaluable memoir byImschenetsky, f/runerfs Arcliiv derMathematik undPhyaik,t.LTV.;andinthememoir by Graindorge abeady (223) quoted. Full references toother authorities are tobefound there. MISCELLANEOUS EXAMPLES. 1.Prove thattheintegraloftheequation (* asgivenbyMonge's method is wherey+xistobesubstituted foraafter integration and/andFare arbitraryfunctions. Hence solve theequation 2.SolvubyMonge's method theequations: (i) (ii) (iii) (iv) (v) (vi) (r-8)a>=(t-8)y; (yii)aA--^-2^+2=0; (viii) (r-s')y+(8-t)a;+q-p=0; (is) o>r+^-a!)8-yt=q-p. 418 MISCELLANEOUS *r 3.Solve theequation r+t=Zs, anddetermine thearbitrary functions 4)y theconditions thatbs=yzwhenx=Qandaz=aP wheny0./ 4.Integrate theequation r__t__p_ _ op ifiy? ifl'3 u andobtain afirstintegraloftheequation y 5.Investigate asolution oftheequation rt-s^O, subjecttotheconditionqz=aP(1+.pa ),intheform 6.Integratetheequation having giventhatpyqx=Q ;andshew thataparticular solution is =Qcosh- . Integratealsotheequation {(l+^*-2pJ + anddisouas thenature ofthesolution 7.Solve theequations: (i)e^(r-p}=^(t-q); (ii) (ui)xr+ajya+yq=Q; (iv) (v)2ar-2+323=0; (vi)x(r-o?t}=2p. 8.Prove thattheonlyrealsolution ofthesimultaneousequations = s 9.Prove thattheonlyreal solutions whichsimultaneously satisfythe equations r+t=Za] arecomprisedin z=Jx*(a+ccosa)+cxysina+}$*(a-acosa) where ca=a2+63anda,/3,y,8arearbitrary parameters. EXAMPLES. 419 % 10.Obtain anintermediary integralof pqr=8(l+pr ), andshew that itsgeneral integralisobtained byeliminating abetween the equations where$and/arearbitrary. (Serret, andGraindorge.) 11. Integrate theequations: (i) (ii)(xp+yq)(rt- (iii) Alsosolve, bychanging theindependent variables to|and17where .^= andx=> and,bychangingtheindependentvariables toand77where #=e^+17and 12. Integratetheequations: 3%23* &z (Gregory.) 13.Find thesurface whose equationsatisfies =0 andwhose traceontheplaneofxyisthehyperbola xy=a?. 14.Integrate thesimultaneous equations 3/3a ." 15.Shew thatthesimultaneous equations rt+c(r+t)=Q, pq+cf(py-qx)=Q, representaseries ofcoaxal paraboloidswhich cutanyfixed plane perpen- dicular tothe RTfiainaseries ofsimilar conies theratio ofwhose ax.es is 420 MISCELLANEOUS *r 16.Shew thattheequation inwhich(7,n,Karefunctions of#,y,zandqcanbeintegratedif andobtain theintegral. Hence obtain theintegralof {(x+yz)s-ypq} (x+y)=qy(1-z) intheform A (Imschenetsky, andGraindorge.) 17.Obtain asolution oftheequation (fiu ffill B8!* Sic83y233a inaseries ofascending powersofx. (Lagrauge.) Solve theequation ,r-9_-_a , ay*J oyoz azz discussinginparticularthecase inwhich thediscriminant oftheleft-hand side iszero. 1^?Verifythatthepartialdifferential equation isintegrableinfinite terms,if6(2j+l)=2i where iisapositive integer. Solve also * (Legendre.) 19.Shew thatthecomplete integralof 1^_9%2&M a*a^2~ 9r3+r^ r beinganinteger) maybeexhibited intheform r(r r where and^arearbitrary functions; andobtain intheform ofadefinite integral thecompletesolution of EXAMPLES. 421 20.Obtain asadefinite integral thesolution of 2,x+y 21.Obtain asolution oftheequation _ Tt~ag? intheform 'roo /*Qo Trti= I Ia"1*-*1 (3+$auvk) dudv. Ja>J-co 22.Change thedependentvariable from etoyintheequation andhence obtain thesolution oftheequationintheform 23.Shew that ifthere befivefunctions z^z.2, ,z^zbeach ofwhich satisfies theequations where thea'sand 6'sarefunctions of#andyalone, thenbetween them there isalinear relation with constant coefficients oftheform If,inaddition, anyfour ofthem aszuz^z&ztbesuch astosatisfy identically theequation *U *2>*4=0: p then there isalsoarelation oftheform Cft+0&+0aZ a+C44=0. (AppeU.) 24.Shew thatthefunction ^(a, ft-y, fl,c, ,y)givenbytheseries n(a+m+n-l) n03+i-l)n(y+n-*l)n(fl-l)n(e-l)^ ' thesummation extendingforallintegralvalues ofm,andnfrom zero to infinity,satisfies thetwoequations (X-s )r-xya+{6-(a+(9+1)x}p-&yq-a|3z=0, (y-ya )*-a#+{-(a+y+1)y}-yap-aye=0. Hence shew thatF(a, 8+c,-c, 0,e,s,y)isasolution of cbeinganarbitraryconstant. (Appell.) 422 MISCELLANEOUS EXAMPLES. 25. Ifthere bethree functionsz^z^essatisfying h(ft~ft)+ga(Pa~ ffi)+3s(ft~ ft)= where the0*3,6'aand c*sarefunctions ofxandy,then there exists between these functions alinear relation with constant coefficients. (Appell.) 26.Shew thattheintegraloftheequation ma7>bydifferentiation, beconnected withthat of s+asyp+(k+n)yz=0, kbeing aconstant andnbeing aninteger. Hence solve theformer equationinthecasewhen kisanegative iutoger. Obtain thesolution when Teisapositive integer. (Tanner.) 27.Obtain thesolution of intheforms=ef where and^arearbitrary functions. Henceintegrate s= Integrate also mtheform(Liouville.) (Tanner.) sin2TO(/*+/) where TZisaconstant, /"(a-)=0 (a?)5(j:)and/'(y)=^ (y)x(y)^^ arearbitrary. (R.Russell.) 28. Integrate byAmpere's method theequations (i)u+ (iii)wyx, (Imschenetsky. ) RAMAN BE8EMIOH WITITUTt BANGALORE6 Glut INDEX. (The figures refer tothepages.) Abel, 249. Amp&re'a method ofsolving thegene- ralised form ofMonge'a equation, 406410. BesseTs equation, 169168; derivable from Legendre'e equa- tion, 169. Bour, 347. Oauohys method ofintegrating Eater's equation, 241. Oayley, 86,92,213, 248. Oharpit's method ofintegration of partial differential equationsofthe first order intwoindependent varia- bles,317324. Olairaut'sequation, 27,312. Classification oftheintegrals ofapar- tialdifferential equation, 287299; every integralisincluded inone ofthethree classes, 291. Complementary Function, 49,6265, 6G,384, 389. Complete Integral ofapartialdiffer- ential equation, 288,866. Cuspidal Locus, 33. Darboux, 86,297. Definite Integrals, solution oflinear equation whose coefficients are of first degree inindependent variable bymeansof,217 223; .. proposition relating tosolution ofgeneral equation bymeans of,223227; solution ofapartial differential equation in,397. Degree, definition of,8. Depressionoforder ofequation when oneormore particular integrals are known, 60,115; when onevariable isabsent, 77. Duality between partial differential equa- tions, analytical, 813,876; corresponds togeometrical prin- ciple ofduality, 316. Envelope Locus, 33;theonly Singular Solution, 36. Equationoffirst order and firstdegree hasonlyoneindependent 'primi&ve, 16. . Equations giving relation between dif- ferential coefficients, 74 76.Equivalence oflinear equations ofsecond order, conditions for,96. Euler, 284, 360. Euler's equation, 289 243; generalisation of,243 249. Exact equations, 8286. Ferrers, 169. FirstIntegrals, definition of,9;num- berofindependent, belongingto equa- tionofnchorder, 9. Functions, conditions forrelations be- tween, 11. Gauss, 186, 212. Gauss'snfunction, 166, Ifil, 198. General Integral ofapartial differential equation, 291. Generalisation ofanyintegral ofapartial differential equation containing con- stants, 410 415. Glaisher,J.W.L.,89,176,178. Goursat, 213. Graindorge, 342,417. Hankel, 161, 167. Heine, 159, 169, 170. Hicks, 153. Homogeneous ordinary equations offirst order, 20; linear ofnltlorder, 66; ingeneral, 79; partial equations, 392. Hypergeometrio Series, definition of,185; differential equation satisfiedby, 187; particular solutions ofthisequa- tion,189194; relations between these solutions, 194-203; oaseswhenexpressible inafinite / form,204212; asadefinite integral, 230. Imaohenetsky, 842,411,417. IndependenceofParticular Integrals of generallinear equation, conditions for,110. Intermediary integral, 856. Invariant ofcoefficients oflinear equa- tion ofsecond order, 89. Jaoobi, 92,213,284, 249,942. J.aoobi'Sjmeth.od ofintegratingthegene- ralised form otEuler's Lqnation,243. 424 INDEX. Jacobi's method fordieintegrationof thegeneral partialdifferential ofthe firstorder innindependent variables, 325842. Kummer, 92,218. Lagrange, 92,301, 317,411. Lagrange'slinear partialdifferential equation, 299303; generalised form, 304. Laplace's transformation ofthelinear partialdifferential equationofthe second order,877382. Legendre,860. Legendre's equation, 143 169. Linear equationwith constant coeffi- cient?, ordinary, Chap.m.; partial, 383393. Lobatto, 234. Lommel, 170, 176. Malet, 90. Mansion,342. Monge's form ofsolution oftotal diffe- rential equations,255. Monge's method ofintegrating the equationofthesecond order which islinear inthepartialdifferential co- efficients, 868-^871. Motion ofparticle under central force, integration ofequations of,278." Neumann, 170. Nodal Locus, 33. NormaJ form oflinear equationofsecond order, 90; ofequation ofhypergeometric series, 188. Order, definitionof,8. Particular Integral, 49,5766, 67,386, 391. Petzval, 234. Poisson's method foraform ofhomo- geneous partial equation, 882. Primitive, definitionof,8.. Quotient oftwo solutions* oflinear equation ofsecondorder, equation satisfiedby,92. Rayleigh, 169. Relation betweenlinearly independent solutions ofadifferential equation, 99,112,155, 168, 201. Eiccati'sequation, 170176j reducible toBeasel's equation, 173. Bichelot, 248. Eiohelot's method ofintegrating Euler's equation, 289.Riemann, 400. !" Routh, 170, 342. Sohwarz, 92,204, 213. Sehwarzian Derivative, 92,204212. Series, possibilityofintegration in,182; form ofsolution when avanish- ingfactor occurs inthedenomi- nator ofacoefficient, 189; formwhen such afactor occurs inthenumerator, 141; integration ofpartial equations in,894396, 401405. Simultaneous equations (ordinary), linear* with constant coefficients," 265272; with variable coefficients, 272 278. Simultaneous partial differential equa- tions inonedependent variable, 347 352. Singular Solutions ofordinary equa- tions offirstorder,3039. Singular Integral ofapartialdiffer- entialequation, 290 ; derived from thedifferential equa- tion, 296. Solution ofordinary equation, what is tobeconsidered a,6. Species, definitionof,7. Spitzer, 234. * Standard Forms ofordinary equations offirst order,1680; ofpartial differential equations offirstorder,806312; they areparticular oases inwhioh Oharpit'e method(q.v.)proves effective, 822 324. Sturm, 170. Symbolic Operations, 48 48,384, 395, 399. Symbolical method forpartial equations duetoLaplace andPoisson, 898.* Symbolical Solutions, 176. Tao-Locus, 35,298. Thomson, SirWilliam, 108. Todhunter, 169, 170. Total differential equations, which are linear,249257; theyseparate intotwoclasses, 266; geometrical interpretation of linear equations with three variables, 268261; case of7ivariables, 261; equations whioh arenot linear, 263. Trajectories, general, 119; orthogonal, 120. Variation ofParameters, SB,112, 116, 411.<* IWUUN fiQHiMOH INSTITUTEUNTVBBBITY PRESS.