Piaggio-Elementary-Treatise-on-Differential-Equations

PAGES MISSING WITHIN THE BOOKONLY <s^ -w ^ [<OU 164828 ^>Qi ;o BELL'S MATHEMATICAL SERIES ADVANCED SECTION General Editor: WILLIAM P.MILNE, M.A., D.Sc. FORMERLY PROFESSOR otMATHEMATICS, LEEDS UNIVERSITY ANELEMENTARY TREATISE ONDIFFERENTIAL EQUATIONS AND THEIR APPLICATIONS ANELEMENTARY TREATISE ON DIFFERENTIAL EQUATIONS ANDTHEIR APPLICATIONS BY H.T.H.PIAGGIO, M.A., D.Sc. PROFESSOR OFMATHEMATICS, UNIVERSITY COLLEGE, NOTTINGHAM rOSWHKT YSENIOR SCHOLAR OFST.JOHN'S COLLEGE, CAMBRIDGE LONDON G.BELLAND SONS, LTD, 1949 First published May 1920. Reprinted 1921, 1924, 1925, 1926 Revised andEnlarged Edition 192* Xeprinted 1929, 1931, 1933, 1937, 19:59, 1940,1911, 1942, 1943, 1944, 1945, 1946, 19iS,1949. PRINTED INGREAT BRITAIN BYROBERT MACLKHOSE AND CO.LTD. THEUNIVERSITY PRESS, GLASC.OW PREFACE "THETheoryofDifferentialEquations,"saidSophus Lie," isthe mostimportantbranch ofmodern mathematics." Thesubject may beconsidered tooccupyacentralpositionfrom which different lines ofdevelopment extend inmanydirections. Ifwetravel along thepurely analytical path,wearesoon ledtodiscuss Infinite Series, Existence Theorems andtheTheoryofFunctions. Another leads ustotheDifferential GeometryofCurves andSurfaces. Between thetwo liesthepathfirstdiscovered byLie,leadingtocontinuous groupsoftransformation and theirgeometrical interpretation. Diverginginanother direction, weareledtothestudyofmechanical andelectrical vibrations ofallkinds andtheimportant phenomenon ofresonance. Certainpartialdifferential equationsform thestart- ingpointforthestudyoftheconduction ofheat, thetransmission ofelectric waves, andmanyother branches ofphysics. Physical Chemistry, with itslawofmass-action,islargelyconcerned with certain differentialequations. Theobjectofthisbook istogiveanaccount ofthecentral partsofthesubjectinassimpleaform aspossible,suitable for those withnoprevious knowledgeofit,andyetatthesame time topointoutthedifferent directions inwhich itmaybedeveloped. Thegreater partofthetextandtheexamplesinthebodyofit willbefound very easy. Theonlyprevious knowledgeassumed is that oftheelements ofthedifferential andintegralcalculus anda little coordinategeometry.Themiscellaneous examplesattheend ofthevariouschaptersareslightlyharder. Theycontain several theorems ofminorimportance,with hints thatshould besufficient toenable thestudent tosolve them. Theyalsocontain geometrical andphysical applications, butgreatcarehasbeen taken tostate thequestions msuchawaythatnoknowledgeofphysicsisrequired. Forinstance, onequestionasks forasolution ofacertainpartial VI PREFACE differential equationinterms ofcertain constants and variables. Thismayberegardedasapieceofpure mathematics, but itis immediatelyfollowed byanotepointingoutthatthework refers toawell-known experimentinheat,andgivingthephysical meaning oftheconstants andvariables concerned.Finally,attheend of thebook isgiven asetof115examplesofmuchgreater difficulty, most ofwhich aretaken fromuniversity examinationpapers. [I have tothank theUniversities ofLondon, Sheffield andWales, and theSyndicsoftheCambridge University Press fortheir kindper- mission inallowing metousethese.] Thebook covers thecourse indifferentialequations requiredfortheLondon B.Sc.Honours or Schedule AoftheCambridge MathematicalTripos,Part II.,and alsoincludes some oftheworkrequiredfortheLondon M.Sc. or Schedule BoftheMathematicalTripos. Anappendix gives sugges- tions forfurtherreading. Thenumber ofexamples,bothworked andunworked, isvery large, andtheanswers totheunworked ones aregivenattheend ofthebook. Afewspecial pointsmaybementioned. Thegraphical method inChapterI.(based ontheMS.kindlylentmebyDr.Brodetsky ofapaper heread before theMathematical Association, andona somewhat similar paper byProf.Takeo Wada) hasnotappeared before inanytext-book. Thechapter dealingwith numerical integrationdeals with thesubjectrather morefully than usual. Itischiefly devoted tothemethods ofRunge andPicard, but it alsogivesanaccount ofanewmethod duetothepresentwriter. Thechapteronlinear differentialequationswith constant co- efficients avoids theunsatisfactory proofs involving"infinite con- stants." Italsopointsoutthattheuseoftheoperator Dinfinding particular integrals requires morejustification than isusually given. Themethod hereadoptedisatfirst tousetheoperator boldly and obtain aresult, andthentoverifythisresultbydirect differentiation. Thischapterisfollowedimmediately byoneonSimplePartial Differential Equations (based onRiemann's "Partielle Differential gleichungen ").Themethods givenareanobvious extension of those intheprevious chapter, andthey areofsuchgreat physical importancethat itseems apitytodeferthem until thelaterportions ofthebook, which ischiefly devoted tomuchmore difficultsubjects. InthesectionsdealingwithLagrange'slinearpartial differential equations, twoexamples have been taken from M.J.M.HilTf recentpapertoillustrate hismethods ofobtaining special integrals PREFACE vii Indealingwith solution inseries, great prominencehasbeen giventothemethod ofFrobenius. Onechapterisdevoted tothe useofthemethod inworkingactualexamples.This isfollowed byamuch harderchapter, justifyingtheassumptions made and dealingwith the difficultquestionsofconvergenceinvolved. An effort hasbeenmade tostate very clearly anddefinitely where the difficulty lies,andwhat arethegeneralideas ofthesomewhat complicated proofs.Itisacommonexperiencethatmanystudents when firstfaced byalong" epsilon-proof"aresobewildered by thedetails thattheyhaveverylittle idea ofthegeneral trend, Ihave tothank Mr.S.Pollard, B.A., ofTrinity College, Cambridge, forhisvaluablehelpwith thischapter.This isthemostadvanced portionofthebook, and,unlike therestofit,requiresalittleknow- ledgeofinfinite series. However, references tostandard text-books havebeengivenforeverysuchtheorem used. Ihave tothank Prof.W.P.Milne, thegeneraleditor ofBell's Mathematical Series, forhiscontinual encouragement andcriticism, andmycolleaguesMr. J.Marshall, M.A., B.Sc., andMissH.M. Browning, M.Sc., fortheir work inverifyingtheexamplesand drawingthediagrams. Ishallbevery gratefulforanycorrections orsuggestions from thosewhousethebook. H.T.H.PIAGGIO. UNIVERSITY COLLEGE, NOTTINGHAM, February,1920. PREFACifi TOTHEREVISED ANDENLARGED EDITION THIS edition contains alongnewchapterofasupplementary character, dealingwith difficulties inthetheoryofsingular solutions, andsome little-known ideas about discriminant-loci regardedas boundaries;Riccati's equation;twoadditional methods fortotal differentialequations (Mayer's general method, andtheuseofan integratingfactor forhomogeneous equations) ;solutions inseries oflinear differentialequationsofthesecond order (Fuchs' theorem, ordinary andsingular points, equationsofFuchsiantype,charac- teristic index, normal andsubnormalintegrals);someequationsof MathematicalPhysics (particularlytheequationofvibrating strings andthethree-dimensional Waveequation);andapproxiinatf numerical solution (Adams' method andsome recent workbj Remes). Theotherpartsofthebookhavebeen revised, andafe\* more,examples added. References havebeen altered whennecessary. Iamdeeply indebted toseveral friends fortheir valuablehelp andadvice, particularlytoMr.H.B.Mitchell, formerlyProfessor atColumbiaUniversity, NewYork, Prof. E.H.Neville ofReading University, andmycolleague, Mr.F.Underwood. H.T.H.PIAGGIO. May 1928. NOTE TOTHENINTH (1933)IMPRESSION FORtheconvenience ofphysicsstudents andothers whorequire a simpletreatment oftheequationofvibrating strings, twoshort noteshavebeenadded(pp.61and256). Themethod ofparameters (p.256)forcertainpartialdifferentialequationsisanextension ot theusual methods fortwostandard forms. Thenewexamples on Lagrange'slinearequation (p.161) include thedetermination of particular integrals representingsurfaces thatpassthrough given curves. There aresome alternative methods forsimultaneous equationsonp.48,andminorchangeselsewhere. H.T.H.P May 1933. viii CONTENTS P401 HISTORICAL INTRODUCTION--., xvii CHAPTER I INTRODUCTION ANDDEFINITIONS. ELIMINATION. GRAPHICAL REPRESENTATION AKT. 1-3.Introduction anddefinitions- 1 4-6.Formation ofdifferential equations byelimination- - 2 7-8.Complete Primitives, Particular Integrals, and Singular Solutions 4 9.Brodetsky andWada's method ofgraphical representation- 5 10.Ordinary andSingular points7 Miscellaneous Examples onChapterI- -10 CHAPTER II EQUATIONS OFTHEFIRSTORDER ANDFIRSTDEGREE 11.Typestobeconsidered 12 12.Exact equations- - - - * - - - -12 13.Integratingfactors 13 14.Variables separate13 15-17. Homogeneous equationsofthefirstorder anddegree,- -14 18-21. Linear equationsofthefirstorderanddegree 16 22.Geometrical problems. Orthogonal trajectories 19 Miscellaneous Examples onChapterII- - - -22 CHAPTER III LINEAR EQUATIONS WITHCONSTANT COEFFICIENTS 23.Typetobeconsidered 26 24.Equationsofthe firstorder -25 iz ; CONTENTS ART. FAG1 25.Equationsofthesecond order 25 26.Modification when theauxiliary equation hasimaginaryor complexroots 26 27.Thecase ofequal roots 27 28.Extension tohigherorders 27 29.TheComplementary Function andtheParticularIntegral-29 30-33. Propertiesoftheoperator D 30 34.Complementary Function when theauxiliary equation has repeated roots 32 35-38. Symbolical methods offinding theParticularIntegral. Ten- tative methods andtheverification oftheresults they give 33 39.Thehomogeneouslinear equation 40 40.Simultaneous linear equations 42 Miscellaneous Examples onChapterIII.(with notes on mechanical and electricalinterpretations,freeand forced vibrations andthephenomenon ofresonance) 43 CHAPTER IV SIMPLE PARTIAL DIFFERENTIAL EQUATIONS 41.Physical originofequationstobeconsidered...4.9 42-43. Elimination ofarbitrary functions andconstants * -49 44.Specialdifficulties ofpartialdifferential equations- - -51 45-46. Particular solutions. Initial andboundary conditions 52 47-48. Fourier's Half-RangeSeries 54 49-50. ApplicationofFourier's Series informing solutionssatisfying given boundaryconditions 56 Miscellaneous Examples onChapter IV.(with notes on theconduction ofheat, thetransmission ofelectric waves andthediffusion ofdissolvedsalts) 57 CHAPTER V EQUATIONS OFTHEFIRST ORDER, BUTNOTOFTHE FIRSTDEGREE 51.Typestobeconsidered........62 52.Equationssolvable forp....-- 62 53.Equationssolvable fory 63 54.Equationssolvable forx 64 CONTENTS CHAPTER VI SINGULAR SOLUTIONS ART. PAO1 55.Theenvelope givesasingular solution 65 66-58. Thec-discriminant contains theenvelope (once), thenode- locus(twice), andthecusp-locus (three times) 66 69-64. The^-discriminant contains theenvelope (once), thetac-locus (twice), andthecusp-locus (once) 71 65.Examples oftheidentification ofloci,using bothdiscriminants 75 66-67. Clairaut's form 76 Miscellaneous Examples onChapter VI- *73 CHAPTER VII MISCELLANEOUS METHODS FOREQUATIONS OFTHE SECOND ANDHIGHER ORDERS 68.Typestobeconsidered- - - - - - - 81 69-70. yorxabsent 82 71-73. Homogeneous equations-...*.. 33 74.Anequation occurringinDynamics 86 75.Factorisation oftheoperator 86 76-77. Oneintegral belongingtothecomplementary function known 87 78-80. Variation ofParameters 88 81.Comparisonofthedifferent methods 90 Miscellaneous Examples onChapterVII.(introducing the Normal form, theInvariant ofanequation, andthe Schwarzian Derivative)- - -91 CHAPTER VIII NUMERICAL APPROXIMATIONS TOTHESOLUTION OF DIFFERENTIAL EQUATIONS 82.Methods tobeconsidered 94 83-84. Picard's method ofintegratingsuccessive approximations-94 85.Numerical approximation direct from thedifferential equa- tion. Simple methodssuggested bygeometry- - -97 86-87. Runge's method 99 88.Extension tosimultaneous equations- -103 89.Methods ofHeun andKutta 104 90-93. Method ofthepresent writer, with limits fortheerror-105 xii CONTENTS CHAPTER IX SOLUTION INSERIES. METHOD OFFROBENIUS ART. FAGB 94.Frobenius* form oftrial solution. The indicial equation-109 95.Case I.Roots ofindicial equation unequal anddiffering bya quantity notaninteger 110 96.Connection between theregion ofconvergenceoftheseries andthesingularitiesofthecoefficients inthedifferential equation 112 97.Case II.Roots ofindicial equation equal- - 112 98.Case III. Roots ofindicial equation differing byaninteger, making acoefficient infinite 114 99.Case IV.Roots ofindicial equation differing byaninteger, making acoefficient indeterminate 116 100.Some cases where themethod fails.Noregular integrals-117 Miscellaneous Examples onChapter IX.(with notes on thehypergeometricseries and itstwenty-foursolu- tions) 119 CHAPTER X EXISTENCE THEOREMS OFPICARD, CAUCHY, AND FROBENIUS 101.Nature oftheproblem121 102. Picard's method ofsuccessive approximation122 103-105. Cauchy's method 124 106-110. Frobenius' method. Differentiation ofaninfinite series with respecttoaparameter- - - - - 127 CHAPTER XI ORDINARY DIFFERENTIAL EQUATIONS WITHTHREE VARIABLES ANDTHECORRESPONDING CURVES ANDSURFACES 111.Theequationsofthischapter express propertiesofcurves and surfaces 133 112.Thesimultaneous equations dx/P=dy/Q=dz/R- 133 113.Useofmultipliers135 114.Asecond integral found bythehelpofthe first- - 136 115.General andspecial integrals- - '37 CONTENTS xiii ART. PAOI 116.Geometricalinterpretationoftheequation Pdx+Qdy +Rdz=Q . -137 117.Method ofintegrationofthisequation when itisintegrable-138 118-119. Necessary and sufficient condition that suchanequation should beintegrable 139 120.Geometricalsignificance ofthenon-integrable equation-142 Miscellaneous Examples onChapter XI- - - 143 CHAPTER XII PARTIAL DIFFERENTIAL EQUATIONS OFTHEFIRST ORDER. PARTICULAR METHODS 121-122. Equationsofthischapterofgeometricalinterest- - -146 123.Lagrange'slinear equation and itsgeometrical interpretation 147 124.Analyticalverification ofthegeneral integral- - -149 125.Special integrals. ExamplesofM.J.M.Hill's methods of obtaining them 150 126-127, Thelinear equation withnindependentvariables- - -151 128-129. Non-linearequations. Standard I.Onlypandqpresent-153 130.Standard II.Only p,q,andzpresent153 131.Standard III.f(x,p)=F(y, q)154 132.Standard IV. Partial differential equations analogousto Clairaut's form 154 133-135.Singular andGeneralintegrals and their geometrical signifi- cance. Characteristics... ---155 136. Peculiarities ofthelinear equation- - - 158 Miscellaneous Examples onChapterXII. (with anoteon thePrincipleofDuality)160 CHAPTER XIII PARTIAL DIFFERENTIAL EQUATIONS OFTHEFIRST ORDER. GENERAL METHODS 137.Methods tobediscussed - - * -162 138139.Charpit's method 162 140-141. Three ormore independentvariables. Jacobi's method 166 142.Simultaneouspartialdifferential equations- - 168 Miscellaneous Examples onChapter XIII- *170 xiv CONTENTS CHAPTER XIV PARTIAL DIFFERENTIAL EQUATIONS OFTHESECOND ANDHIGHER ORDERS ART. PA01 143.Typestobeconsidered 172 144.Equations thatcanbeintegrated byinspection. Determina- tion ofarbitraryfunctions bygeometricalconditions-172 145-151. Linearpartialdifferential equationswithconstant coefficients 173 152-153. Examplesinelimination, introductorytoMonge's methods-179 154.Monge's method ofintegrating Rr+Ss+Tt=V- -181 165.Monge's method ofintegrating Rr+Ss+Tt+U(rt-s*)=V-183 156-157. Formation ofIntermediate Integrals183 158.FurtherintegrationofIntermediateIntegrals- - 186 Miscellaneous Examples onChapter XIV. (with notes on thevibrations ofstrings, bars,andmembranes, and onpotential)188 CHAPTER XV MISCELLANEOUS METHODS 159.Methods tobediscussed 191 160.Some difficulties inthetheoryofsingular solutions 192 161.Discriminants, Particular Solutions, andBoundaries - 194 162. Riccati's equation- 201 163.Reduction toalinearequationofthesecond order- -201 164.The cross-ratio ofanyfour particular integralsofaRiccati's equationisindependentofx 202 265.Method ofsolution when three particular integralsareknown 202 166.Method ofsolution whentwoparticular integrals areknown-202 167.Method ofsolution when oneparticular integralisknown-203 168.Twomethods ofintegrating thetotal differential equations Pdx+Qdy+Rdz=Q... -205 169.Integrating factor forhomogeneous equations- - 205 170.Mayer's method 206 171.Linear differential equationsofthesecond order... 208 172.Regular integrals 208 173.Fuchs' theorem- - - -211 174.Ordinary andsingular points...... 212 175.EquationsofFuchsian type 213 176. Characteristic index 214 177.Normal andsubnormalintegrals-*- 215 178.Theequation ofvibrating strings- * 218 CONTENTS XV ABT. PAOB 179. Particular solutions oftheWave equation219 180. Poisson's (orLiouville's) generalsolution 220 181.Other differential equationsofMathematical Physics222 182.Numerical approximation. Adams' method 224 183.Remes' extension ofthemethod ofArts. 90-93... 227 APPENDIX A Necessary andsufficient condition thattheequation should beexact.-....,. 229 APPENDIX B Anequation withnospecial integrals 230 APPENDIX C The equation found byJacobi's method ofArt. 140 is always integrable........ 231 APPENDIX D Suggestionsforfurther reading- 232 MISCELLANEOUS EXAMPLES ONTHEWHOLE BOOK (with notes onsolution bydefiniteintegrals, asymptotic series, theWronskian, Jacobi's last multiplier,finite difference equations, Hamilton's dynamical equations,Foucault's pendulum, andtheperihelion ofMercury)... -233 ANSWERS TOTHEEXAMPLES- i NOTE ONALTERNATIVE ANSWERS xxii IDEX** XXV HISTORICAL INTRODUCTION THEstudyofDifferential Equations began verysoon after the invention oftheDifferential andIntegral Calculus, towhich it forms anaturalsequel. Newton in1676 solved adifferential equation bytheuseofaninfinite series, onlyelevenyearsafter hisdiscoveryofthefluxional form ofthedifferential calculus in 1665. Butthese results were notpublisheduntil 1693, thesame yearinwhich adifferential equationoccurred forthe firsttime in thework ofLeibniz*(whose account ofthedifferential calculus waspublishedin1684). Inthenext fewyears progress wasrapid. In1694-97 John Bernoullifexplainedthemethod of" SeparatingtheVariables/' and heshowed how toreduce ahomogeneousdifferential equationof the firstorder tooneinwhich thevariables wereseparable. He appliedthese methods toproblemsonorthogonal trajectories. He and hisbrother Jacob tt(afterwhom"Bernoulli's Equation" is named) succeeded inreducingalargenumber ofdifferentialequa- tions toformstheycould solve.Integrating Factors wereprobably discovered byEuler (1734) and(independentlyofhim)byFontaine and Clairaut, though some attribute them toLeibniz.Singular Solutions, noticed byLeibniz (1694) andBrookTaylor (1715), are generallyassociated with thename ofClairaut (1734). Thegeo- metricalinterpretationwasgiven byLagrangein1774, butthe theoryinitspresentformwasnotgivenuntilmuch laterbyCayley (1872) andM.J.M.Hill(1888). The firstmethods ofsolvingdifferential equationsofthesecond orhigherorders with constant coefficients were due toEuler. D'Alembert dealt with the'casewhen theauxiliary equation had equalroots. Some ofthesymbolicalmethods offindingthepar- ticularintegralwere notgivenuntil about ahundredyearslater byLobatto (1837) andBoole (1859). The firstpartialdifferential equation"tobenoticed wasthat givingtheform ofavibrating string.Thisequation, which isof thesecond order, wasdiscussed byEuler andD'Alembert in1747. Lagrange completedthesolution ofthisequation, andhealso *Also spelt Leibnitz. tAJ8spelt Bernoulli, ftAlsoknown asJames. HISTORICAL INTRODUCTION dealt, in-aseries ofmemoirs from 1772 to1785, withpartialdif- ferentialequationsofthe first order. Hegave thegeneral integral ofthelinearequation, and classified thedifferent kinds ofintegrals possible when theequationisnotlinear. These theories stillremain inanunfinished state;contributions havebeenmaderecently bfChrystal (1892) andHill(1917). Other methods fordealingwithpartialdifferentialequationsofthe first order weregiven byCharpit (1784) andJacobi(1836). Forhigher orders themostimportant investigationsarethose ofLaplace (1773), Monge (1784), Ampere (1814), andDarboux(1870). Byabout 1800thesubjectofdifferentialequationsinitsoriginal aspect, namely thesolution inaforminvolving onlyafinitenumber ofknown functions(ortheirintegrals), wasinmuch thesame state asitisto-day. Atfirstmathematicians hadhopedtosolveevery differential equationinthisway,buttheir effortsprovedasfruitless asthose ofmathematicians ofanearlier date tosolve thegeneral algebraic equationofthe fifth orhigher degree. Thesubject now became transformed, becoming closelyallied totheTheoryof Functions. Cauchyin1823provedthattheinfinite series obtained from adifferentialequation wasconvergent, andsoreallydid define afunctionsatisfyingtheequation. Questions ofconvergency (for^which Cauchy wasthe first togive tests) areveryprominent jnalltheinvestigationsofthissecondperiodofthestudyofdif- ferential equations. Unfortunatelythismakes thesubject very abstract and difficult forthestudent tograsp. Inthe firstperiod theequationswerenotonlysimplerinthemselves, butwere studied inclose connection withmechanics andphysics, which indeed were often thestarting point ofthework. Cauchy's investigationswere continued byBriot andBouquet (1856), andanewmethod, that of"Successive Approximations," wasintroduced byPicard(1890). Fuchs (1866) andFrobenius (1873) have studied linearequationsofthesecond andhigher orders with variable coefficients. Lie's TheoryofContinuous Groups (from 1884) hasrevealed aunity underlying apparently disconnected methods. Schwarz, Klein, andGoursat havemade theirwork easier tograsp bytheintroduction ofgraphicalcon- siderations, andarecentpaperbyWada(1917) hasgivenagraphical representationoftheresults ofPicard andPoincare. Runge (1895) andothers have dealt withnumericalapproximations. Further historical notes willbefound inappropriate places throughoutthebook. Formore detailed biographies,seeRouse BalTs Short History ofMathematics. CHAPTER 1 INTRODUCTION ANDDEFINITIONS. ELIMINATION. GRAPHICAL REPRESENTATION 1.Equationssuch as *y_ * (4v ~jT 7^, V/ax7/s/ii/yflx Involvingdifferential coefficients, arecalledDifferential Equations. 2.Differential Equationsarisefrommany problemsinAlgebra, Geometry, Mechanics, Physics, andChemistry.Invariousplaces inthisbookweshallgiveexamplesofthese, including applications toelimination, tangency, curvature, envelopes,oscillations of mechanical systemsand ofelectric currents, bendingofbeams, conduction ofheat, diffusion ofsolvents, velocityofchemical reactions, etc. 3.Definitions. Differential equationswhich involve only one independent variable,* like(1), (2), (3),and(4),arecalled ordinary. Those which involve twoormore independentvariables and partialdifferential coefficients withrespecttothem, such as(5),are calledpartial. *Inequations (1),(2),(3),(4)xistheindependent andythedependentvariable. In(5)aand *arethetwoindependentvariables andythedependent. 2 DIFFERENTIAL EQUATIONS Anequationlike(1),which involves asecond differential co- efficient, butnone ofhigher orders, issaidtobeofthesecond order. (4)isofthefirstorder, (3)and(5)ofthesecond, and(2)ofthethird. Thedegreeofanequationisthedegreeofthehighestdifferential coefficient when theequationhasbeenmade rational andintegral asfarasthedifferential coefficients areconcerned. Thus(1), (2), (4)and(5)areofthe firstdegree. (3)must besquaredtorationalise it.Wethen seethat itisof theseconddegree,as-r~occurssquared. Notice that this definition ofdegreedoesnotrequirexoryto occur rationallyorintegrally. Other definitions willbeintroduced whentheyarerequired. 4.Formation of differential equations byelimination. The problemofelimination willnowbeconsidered, chiefly because it givesusanidea astowhat kind ofsolution adifferentialequation mayhave. We shallgivesome examplesoftheelimination ofarbitrary constants bytheformation ofordinarydifferentialequations. Later (Chap. IV.)weshall seethatpartialdifferentialequations may be formed bytheelimination ofeitherarbitrary constants orarbitrary functions. 5.Examples. (i)Consider x=*A cos(pt-a),theequationofsimple harmonic motion. Letuseliminate thearbitraryconstants Aand a. dx Differentiating,-=-=-pAsin(pt a)at and-jfi=-pzAcos(pt-a)=-p*x. Thus -=-jj-=-p*xistheresultrequired, anequationofthesecond order, whoseinterpretationisthattheacceleration varies asthedistance from theorigin.. (ii)Eliminate pfrom thelastresult. Differentiating again, -^=>-p2-=- . (tt (it __ cPxldx,dzxl/f,11. i..Hence -^37--p2=-JTS\x,(from thelastresult).at9 1at at* \ ffixdxd*x . i-Ti Multiplying up,x .-^-=--^,anequationofthethird order. ELIMINATION 3 (iii)Form thedifferentialequationofallparabolas whose axis is theaxisofx. Such aparabola musthaveanequationoftheform t/8= Differentiating twice, weget (-/)~^ andyj^+(-/)~^which isofthesecond order. Examples forsolution. Eliminate thearbitraryconstants from thefollowing equations: (1)y=Ae**+Be~2*.(2)y=Acos3x+Bsin3x. (3)y-Ae**. (4)y=Ax +A*. (5)Ifxa+t/2=a2 ,provethat-p=--,andinterpretthe result geometrically.x V (6)Prove that foranystraightlinethroughtheorigin-~-r> and interpretthis.xx * UIII (7)Prove that forany straightlinewhatevery4=0. Interpret this.dx 6.Toeliminate narbitrary constants requires (ingeneral) adiffer- ential equation ofthenthorder. The reader willprobably have arrived atthisconclusionalready, from theexamplesofArt. 5. Ifwedifferentiate ntimes anequation containing narbitrarycon- stants, weshall obtain (n+1)equations altogether, fromwhich the nconstants canbeeliminated. Astheresult contains annthdiffer- ential coefficient,itisofthenthorder.* *Theargumentinthetext isthat usually given, buttheadvanced student willnotice someweak points init.Thestatement thatfromany(n+1)equations nquantitiescanbeeliminated, whatever thenatureofthoseequations,istoosweeping.Anexact statement ofthenecessary and sufficient conditions would beextremely complicated.Sometimes lessthan (n+1)equationsarerequired. Anobvious case ia y=(A+B)x,where thetwoarbitrary constants occur insuch awayastobe really equivalenttoone. Alessobvious case isy*=2Axy+Bx*. Thisrepresents two straightlines through theorigin, sayi/=m la;andy^m zx,from each ofwhich weeasily get -=~, ofthe firstinstead ofthesecond order. Thestudent should alsoobtainxdx thisresult bydifferentiating theoriginal equation andeliminating B.This will give 4 DIFFERENTIAL EQUATIONS 7.Themost general solution ofanordinary differential equation of thenthorder contains narbitrary constants. This willprobably seem obvious from theconverse theorem tHat ingeneral narbitrarycon- stants canbeeliminated byadifferentialequationofthenthorder. Butarigorous proofoffersmuchdifficulty. If,however, weassume*thatadifferential equation hasasolution expansibleinaconvergentseries ofascending integral powersof x,wecaneasilyseewhy-thearbitraryconstants areninnumber. Consider, forexample,~=-f-,oforder three.9 r'dx3dx y (r^Assume thaty=a4-^x+#291++an~i+ tinfinity. Then, substitutinginthedifferentialequation, weget 80 08=01, 04=02, B=an_2=On_4=etC. Hence y= ==a-fajsinhx+a2(coshx- 1), containingthree arbitrary constants, a,axanda2. Similar reasoning appliestotheequation -~dx~'*dx'dx2''"'"dx" InDynamics thedifferential equationsareusually ofthesecond d2y order, e.g.-rf+P2y=0,theequationofsimple harmonic motion. Togetasolution withoutarbitrary constants weneedtwocon- ditions, such asthevalue ofyanddy/dtwhen t=0,givingtheinitial displacementandvelocity. 8.Complete Primitive, Particular Integral, Singular Solution. The solution ofadifferential equation containingthe fullnumber of arbitraryconstants iscalled theCompletePrimitive. Anysolution derived from theCompletePrimitive bygiving particularvalues tothese constants iscalled aParticularIntegral. *Thestudent wiU fleeinlater chapters that thisassumptionisnotalways justifiable. GRAPHICAL REPRESENTATION Thus theCompletePrimitive ofrr-|=-p isya-faxsinhx+a%(cosh x-1), ory^c+Oj sinhx+a2cosh x,where c=a-02, oryc+oe35+be~x ywhere a\(a^-fa2)and6 This illustrates thefactthattheCompletePrimitive mayoften bewritten inseveral different (but really equivalent) ways. ThefollowingareParticularIntegrals: y=5sinh x, taking 04=5,c=a2= ; y=6coshx-4, takingaa=6,^=0,0= -4; y=2+ex-3e~*, takingc=2,a=l,6=-3. Inmostequations everysolution canbederived from theCom- plete Primitive bygivingsuitable values tothearbitrary constants. However, insomeexceptionalcasesweshall findasolution, called aSingular Solution, thatcannot bederived inthisway. These will bediscussed inChap. VI. Examples forsolution. Solvebythemethod ofArt.7: (i) g-r- (3)Show thatthemethod fails for-~~^ - v'dxx [logxcannot beexpandedinaMaclaurinseries.] (4)Verify byelimination ofcthatyex+-istheComplete Primitive ofy=x -f1/ .Verifyalsothaty*ixisasolution ofthedifferential equationnotderivable from theCompletePrimitive(i.e.aSingular Solution). Show that theSingularSolution istheenvelopeofthe familyoflinesrepresented bytheCompletePrimitive. Illustrate by agraph. 9.Graphical representation. Weshallnowgivesome examples ofamethod*ofsketching rapidlythegeneral form ofthefamilyof curvesrepresentingtheCompletePrimitive of *DuetoDr.S.Brodetsky andProf.Takeo Wad*. 6 DIFFERENTIAL EQUATIONS wheref(x, y)isafunction ofxandyhavingaperfectlydefinite finite value*forevery pairoffinite values ofxandy. Thecurves ofthefamilyarecalled the characteristics ofthe equation. n /*\ dyEX.(l)J.=X(y-\\ Here Nowacurve has itsconcavity upwards when thesecond differential coefficient ispositive. Hence thecharacteristics willbeconcave up above y=l,andconcave down below this line. Themaximum or minimumpointslieonx=Q,sincedy/dx=Qthere. Thecharacteristics near y1,which isamember ofthefamily,are flatter than those further from it. These considerations show usthatthefamilyisofthegeneral form shown inFig.1. M N Fia. i. Ex.(ii) Here dx Westartbytracing thecurve ofmaxima andminimay-fea=0, andthecurve ofinflexionst/+2e*=0. Consider thecharacteristic through theorigin. Atthispoint both differential coefficients are positive,soasxincreasesyincreases also,andthecurve isconcave upwards.Thisgivesustheright-hand portionofthecharacteristic marked 3inFig.2.Ifwemove tothe leftalongthiswegettothe *Thuaexcludingafunction likey/x,which it*indeterminate whenx0and 0=0. GRAPHICAL REPRESENTATION 7 curve ofminima. Atthepointofintersection thetangentisparallelto Ox. After thisweascendagain,someetingthecurve ofinflexions. Aftercrossingthisthecharacteristic becomes convex upwards.Itstill ascends. Now thefigure shows that ifitcutthecurve ofminima again y Fro. 2. thetangent could notbeparalleltoOx,soitcannot cut itatall,but becomes asymptotictoit. Theother characteristics areofsimilar nature. Examples forsolution. Sketch thecharacteristics of : (2) (3)_ ~dx dy dxy+x*. 10.Singular points. Inallexampleslikethose inthe last article, wegetonecharacteristic; andonly one,through every point dy d2 t/ oftheplane. Bytracingthetwocurves -3-=0and-r\=0wecan easilysketch thesystem. If,however, f(x, y)becomes indeterminate foroneormore points (called singular points),itisoften very difficult tosketch the 8 DIFFERENTIAL EQUATIONS systemintheneighbourhoodofthesepoints. Butthefollowing examples canbetreatedgeometrically. Ingeneral, acomplicated analytical treatment isrequired.* Ex.(i).-=^=^. Here theoriginisasingular point. Thegeo-dx sc metrical meaningoftheequationisthat theradius vector andthe tangent have thesamegradient,which canonlybethecase forstraight FIG. 3. linesthroughtheorigin. Asthenumber ofthese isinfinite, inthiscase aninfinite number ofcharacteristicspassthroughthesingular point. -0 ,... dy x,ydy Jbjx.(u).-=3i.e.-'-r-= 1.dxy xdx Thismeans thattheradius vector andthetangent havegradients FIG. 4. whose productis-1, i.e.that they areperpendicular. The char- acteristics aretherefore circles ofanyradius with theoriginascentre. *Seeapaper," Graphical Solution," byProf,Takeo Wada, Memoirsofth* College ofScience, Kyoto Imperial University, Vol. II.No. 3,July 1917. GRAPHICAL REPRESENTATION 9 Inthiscasethesingular pointmayberegardedasacircle ofzero radius^ thelimitingform ofthecharacteristics near it,butnocharacteristic of finite sizepasses throughit. -,..v dyy-kxEx.(m). /C3^ i- v 'dxx+ky Writing dy/dx**ta>n\[s> y/o?=tan 6,weget tan i.e. tan9-tan \Isf*&___________i!=jif 1-ftan' i.e.tan(6- \js)=k,aconstant. The characteristics arethereforeequiangular spirals,ofwhich the singular point (the origin)isthefocus. FIG. 5. These threesimple examplesillustrate threetypicalcases. Sometimes &finitenumber ofcharacteristicspassthroughasingular point,butanexampleofthiswould betoocomplicatedtogive here.* SeeWada's paper. 10 DIFFERENTIAL EQUATIONS MISCELLANEOUS EXAMPLES ONCHAPTER L Eliminate thearbitraryconstants from thefollowing: (1) y (2) y [Toeliminate A,B,Cfromthefourequations obtained bysuccessive differentiation adeterminant maybeused.] (3) yex(Acosx+Bsinx). (4) yccosh-,(thecatenary).c Find thedifferential equationof (5)Allparabolas whose axes areparalleltotheaxis ofy. (6)Allcircles ofradius a. (7)Allcircles thatpassthroughtheorigin. (8)Allcircles (whatevertheir radii orpositionsintheplane xOy). [Theresult ofEx.6maybeused.] (9)Show thattheresults ofeliminatingafrom (1) andbfrom y**x-~-bx* t...............................(2)cLx areineach case x27-|-2x~+2z/=...... *.....................(3) [The complete primitiveofequation (1)mustsatisfy equation (3), since(3)isderivable from(1). Thisprimitivewillcontain aandalso anarbitraryconstant. Thus itisasolution of(3)containing two constants, both ofwhich arearbitraryasfaras(3)isconcerned, asa does notoccur inthatequation.Infact, itmust bethecomplete primitiveof(3). Similarly thecomplete primitivesof(2)and(3)are thesame. Thus(1)and(2)have acommoncomplete primitive.] (10)Applythemethod ofthelastexampletoprovethat andy~-/-ydx haveacommoncomplete primitive. (11)Assumingthatthe firsttwoequationsofEx.9have acommon complete primitive,find itbyequatingthetwovalues of~intermsdx ofx,y,andtheconstants.Verify that itsatisfiesequation (3)ofEx. 9. (12)Similarly obtain thecommoncomplete primitiveofthetwo equationsofEx.10. MISCELLANEOUS EXAMPLES 11 (13)Prove that allcurves satisfyingthedifferentialequation dx \dx/ dx cuttheaxisofyat45. (1*1)Find theinclination totheaxis ofxatthepoint (1,2)ofthe twocurves whichpassthroughthatpointandsatisfy (15)Prove that theradius oicurvature ofeither ofthecurves of Ex.14atthepoint (1,2)is4. (16)Prove that ingeneral twocurvessatisfyingthe differential equation passthrough any point, butthat these coincide foranypoint ona certainparabola,which istheenvelopeofthecurves ofthesystem. (17)Find thelocus ofapointsijich that thetwocurvesthroughit satisfyingthedifferentialequationofEx.(16)cut(i)orthogonally; (ii)at45. (18)Sketch(byBrodetsky andWada's method) thecharacteristics of (19)Obtain solutions inseries ofascending integral powersofx(as InArt.7)ofthefollowingdifferentialequations (inwhichyandy% denote7and79respectively):dx dx2r (iii)2^-2x^ +20-0; (iv)(l- (v)(x- [Answers: x2x* x* (/p23.8 3.4\ aJ~^-j-+^T-^j-f...)= a^xe~x ;this, containing onlyone 1 ! 2t !o } / arbitrary constant,isnotthecomplete primitive,forthere is another solution notoftheformassumed here (seeChap. IX.) ; (iii)t/^ajX +Ja^2 ; (v) ,v=a(l2+22z+32z2+...); seeArt.97.] CHAPTER II EQUATIONS OFTHEFIRSTORDER ANDFIRSTDEGREE 11.Inthischapter weshall consider equations oftheform =,dx whereMandNarefunctions ofbothxandy. Thisequationisoften written,* moresymmetrically,as Unfortunatelyitisnotpossibletosolve thegeneral equationof thisform interms ofafinitenumber ofknown functions, butwe shall discuss somespecial typesinwhich thiscanbedone. Itisusual toclassifythesetypesas (a)Exactequations; (6)Equationssolvable byseparationofthevariables; (c)Homogeneous equations; (d)Linearequationsofthe firstorder. Themethods ofthischapterarechiefly duetoJohn Bernouilli ofBale (1667-1748), themostinspiring teacher ofhistime, andto hispupil, Leonhard Euler, also ofBale (1707-1783).Eulermade greatcontributions toalgebra, trigonometry, calculus, rigiddynamics, hydrodynamics, astronomy andothersubjects. 12.Exact equations, f Ex.(i).Theexpression ydx+xdyisanexact differential. Thus theequation ydx+xdy=* 0, givingjZ(yz)=0, t.e.yx**c, iscalled anexactequation. *Forarigorous justification oftheuseofthedifferentials<Ja*andAyseeHardy's Pure Mathematics, Art.136[Arts. 154-155 inlatereditions]. tForthenecessary and sufficient condition thatMdx+N(fy=0 should beexact teeAppendixA. 12 EQUATIONS OFFIRSTORDER ANDFIRST DEGREE 13 Ex.(ii).Consider theequationtany.c&c+tanx .dy=Q. This isnotexact asitstands, but ifwemultiply bycos a?cosyit comesBjnycosxdx+sinxcosydy=0, lich isexact. Thesolution is sinysinx**c. 13.Integrating factors. Inthe lastexample cosxcosyis called anintegrating factor,because when theequationismultiplied byitwegetanexactequation which canbeatonceintegrated. There areseveral ruleswhich areusually givenfordetermining integratingfactors inparticularclasses ofequations. These willbe found inthemiscellaneous examplesattheendofthechapter. The proofofthese rulesforms aninteresting exercise, but itisgenerally easier tosolve exampleswithout them. 14.Variables separate. dx Ex.(i).Intheequation=tany.dy,theleft-hand sideinvolvesx xonlyandtheright-handsideyonly,sothevariables areseparate. Integrating, weget logx-logcosy+c, i.e.log(xcos t/)=c, BCOS t/=ec~a,say. Ex.(ii).Tx^Xy' Thevariables arenotseparateatpresent, butthey caneasilybe made so.Multiply bydxanddivide byy.Weget ~-%xdx. y Integrating, logy=x2+c. Ascisarbitrary, wemayputitequaltologa,where aisanother arbitraryconstant. Thus, finally, y^ae^. Examples forsolution. (1)(I2x+5y-d)dx +(5x+2y-4:)dy~Q. (2){cosxtany+cos(x-fy)\dx+{sinxsec2y4-cos(x+y)}dyQ. (3)(secxtanxtany-e*)dx+secxsec2ydy=0. (4)(x+y)(dx~dy)=dx+dy. (5)ydx'-xdy +3x*y*e<*dx=0. (6)ydx-xdy=*Q. (7)(sinx-4-cosx)dy+(cosxsinx)dx=0. (8)J-.V. (9)ydx~xdy*=*xydx.'in *tanxdy=* cotydx. P.D.I. 14 DIFFERENTIAL EQUATIONS 15.Homogeneous equations. Ahomogeneous equationofthe firstorder anddegreeisonewhich canbewritten intheform dx Totestwhether afunction ofxandycanbewritten intheform oftheright-hand side, itisconvenient toput y-=vor 11vx.x Iftheresult isoftheform/(v),i.e. ifthex'sallcancel, the test issatisfied. _, ,.. dii o;2+?/2 . dy 14-v2_.. Jbx.(i).j-=- 3becomes-=.Thisequationishomo- geneous.ax zx dx * Ex.(ii).^-= ,becomes~=a?v3 .This isnothomogeneous. ^dxx2rfxb 16.Method ofsolution. Since ahomogeneous equationcanbe reduced to^=f(v) byputting y=vxontheright-hand side, itis natural totrytheeffect ofthissubstitution ontheleft-hand side also. Asamatter offact, itwillbefound thattheequationcan alwaysbesolved*bythissubstitution(seeEx.10ofthemiscel- laneous setattheendofthischapter). Put y33*^, dy dv t.*-v+x~>(forifyisafunction ofx,soisv). -u *> l+v* Theequation becomes v-fx-=-=- , ^dx 2 i.e.2xdv*=(l+v2-2v)dx. .,, 2dv dx Separatingthevariables, rr^== -\V1jX o Integrating, j-logx+c. y -2 -2-2x 2x But v=-,sor- =v-ly_ly-xx-y x Multiplying byx-y,2x=(x-y)(logx+c). *By"solved"wemean reduced toanordinary integration. Ofcourse,' tbii integral maynotbeexpressibleinterms ofordinary elementary functions. EQUATIONS OFFIRSTORDER ANDFIRSTDEGREE 15 Ex.(ii), (x+y)dy+(x-y)dx-0. Thisgives *-Z.*dxy+x Putting yvx,andproceedingasbefore, weget dvv1 v+x-j-- ?,dxv+1 , dvv1 vf+l i.e. x-j-** --t>=--_. dx v+1 v+1 a-xi i-i (v+l)dv dx Separating thevariables,-- $=, -vdv dv dx **' Integrating,-\log(vz+1)-tan"1 *;logx+c, i.e. 2logx+log (v2+1)+2tan^t;+2c=0, Iogaj2(v2+l)-l-2tan-1v-fa0, putting2c Substitutingforv,log (t/2+x2 )+2tan-1-+a0. 17.Equations reducible tothehomogeneous form. Ex.(i).Theequationy^y- w ^dx isnothomogeneous. Thisexampleissimilar toEx.(ii)ofthelastarticle, except that y-x.,,,y-x+l-isreplaced by-- -.r J-- Nowy-x=andy+x=represent twostraightlinesthrough the origin, Theintersection oft/-aj+l=0and t/-fz+5= iseasily found to be(-2, -3). Putaj==-X'-2; y^Y-3.Thisamounts totaking newaxesparallel totheoldwith(-2,-3)astheneworigin. Then y-x+l~Y-X and y+a;+5=Y+X. Also dxdXand dydY. u<*F^-^TheequationbecomesJv^ ~yv' Asinthelast article, thesolution is f.e. 18 DIFFERENTIAL EQUATIONS ..'axy- Thisequationcannot betreated asthelastexample, because the linesy-z+l=0andy-x+5=*Q areparallel. Astheright-handside isafunction oty-x, tryputting y-x**z, dy dz %*6. -z--1=^= dx dx dz z-+1Theequation becomes 1-f-=-*-, ^dx 2+5 .dz-4 i.e. -r-=a--. dx 2+5 Separatingthevariables, (z+5)dz-4r/a;. Integrating, ^ Substitutingfor2,(y-x)2+10(y- a?)-fSx=2c, i.e.(y- a;)2+10^-2sc a,putting2c a. Examples forsolution. (1)(2x-y)dy~ (2y-x)dx. [Wales. ] (2)(x2-y*)~~ xy. [Sheffield. ]dx \ (8)2&"l+&[Math. Tripos.] (4)xd (6) _ dx3z-4i/-3' * (8)(x+2y)(dx-dy) 18.Linear equations. Theequation ^+Py=6, wherePandQarefunctions ofx(butnotofy),issaidtobelinear ofthe first order. Asimple exampleis+- .y-a^. EQUATIONS OFFIRSTORDER ANDFIRSTDEGREE 1? Ifwemultiplyeach sideofthisby ,itbecomes .. Hence, integrating, xy=Jx*+c. Wehave solved thisexample bytheuseoftheobviousintegrating factor x. 19.Letustrytofindanintegratingfactor inthegeneralcase. IfRissuchafactor, then theleft-hand sideof isthedifferential coefficient ofsomeproduct, andthe firstterm R j-shows thattheproduct must beRy. Put, therefore,R+RPy-Ry^R +y. aR -7- ,mi T*T\ aR Thisgives Hry=y-7- , i.e. Pdx--j>,JK i.e.(pdx=logR, ,9 Thisgivestherule :Tosolve-p-fPy=<?,multiplyeach sideby \Pdx t,which willbeanintegrating factors. 20.Examples. (i)Take theexampleconsidered inArt. 18. -+- .yx.dxxy HereP-, so|P(fo=loga;,and elo****x. x J Thus therulegivesthesameintegratingfactor thatweused before. (ii) HereP=23,|P<fcc=a;2 ,andtheintegratingfactor ise*1 . 18 DIFFERENTIAL EQUATIONS Multiplying oythis, e*2~-+2xe*y-2, Integrating, yef=2sc+c, (iii) Here theintegratingfactor ise8*. Multiplying bythis,e3*4-3e3a ty=e6a , Integrating, ye3*=^e6*-fc, 21.Equations reducible tothelinear form. Ex.(i). xy-~=y*e~x *. Divide byy3 ,soastofreetheright-handsidefromy. rrr 11 dijWeget x.-o--i-;-=e-*\ yydx 11d(\ Putting-==z> 2xz+-j- & 2/2dx This islinear and, infact,issimilar toEx.(ii)ofthelastarticle with instead ofy. Hence thesolution is z*(2x+c)e~**f Thisexampleiaaparticularcase of"Bernoulli'sEquation" wherePandQarefunctions ofx.Jacob Bernouilli orBernoulli of Bale (1654-1705) studied itin1695, EQUATIONS OFFIRSTORDER ANDFIRSTDEGREE 19 Ex.(ii). This isnotlinear asitstands, but ifwemultiply by-=-,weget dx . i.e.dx^-.dyyy This islinear, considering yastheindependentvariable. Proceedingasbefore, wefindtheintegratingfactor tobey2 ,and thesolutionyz=2y*+c, i.e.x=2y8H-cy~I . Examples forsolution. (1)(z+a)|-3y=:(z+a)6 .[Wales.] (2)xcosx~-+y(xsinsc+cosx)l. [Sheffield.] 2-3~(3)a;loga?-^4-y=-2logsc.(4)x2y-x3~==y*cosx. (6) ?/+2d-~y*(x-l). (6)| (7)dx+xdy=e~ysec2 1/dy. 22.Geometrical Problems. Orthogonal Trajectories. We shall now consider somegeometrical problems leadingtodifferential equations. T N FIG. 6. Ex.(i).Find thecurve whose subtangentisconstant. Thesubtangent TN-PNcot^-y~. 20 DIFFERENTIAL EQUATIONS dx . __ Hence -j- ..4. puttingthearbitrary constant cequaltokloga. Ex.(ii).Find thecurve such that itslength between anytwo pointsPQisproportionaltothedifference ofthedistances ofQandP from afixedpoint0. IfwekeepPfixed, thearcQPwillvaryasOQminus aconstant. Usepolar co-ordinates, takingaspoleandOPasinitial line. Then,ifQbe(r,0),wehavej.^-ir But, asshown intreatises ontheCalculus, Hence, inourproblem, r)*=(rae)* +(dr)\ -l)7 Idr-- ,say.ar9J9 givingr(#**,theequiangular spiral. Ex.(iii).Find theOrthogonal Trajectoriesofthefamilyofsemi- cubicalparabolas at/2=a^,where aisavariableparameter. Two families ofcurves aresaidtobeorthogonal trajectories when everymember ofonefamilycutseverymember oftheother atright angles.We firstobtain thedifferential equationofthegiven family by eliminatinga. Differentiating ayzx3 , weget 2ay~^3x^dx whence, bydivision,--^=4- .................................. (1)yctxx ditNow-~=tan\/r,where\/ristheinclination ofthetangenttotheuX axis ofx.Thevalue of^forthetrajectory, say^/,isgiven by tf-^'fcr, i.e.tan\js-cot\f/t i.e.-~~forthegiven familyistobereplaced by--=- forthetrajectory, EQUATIONS OFFIRSTORDER ANDFIRSTDEGREE 21 Makingthischangein(I),weget 2dx 3 afamily ofsimilar andsimilarlysituatedellipses. Ex.(iv).Find thefamilyofcurves that cutthefamilyofspirals r=a0ataconstantanglea. Asbefore, westartbyeliminatinga. Thisgives ~^0. (IT Now ==tan0,where <J>istheangle between thetangent andthe radius vector. If<{>'isthecorresponding angleforthesecond family, tf>'=0a, ,tan<pdbtana *~lqFtan<tanar^I0' puttinginthevalue found fortan <f>andwriting kinstead ofdbtana. Thus, forthesecond family, Thesolution ofthis willbeleftasanexercise forthestudent. Theresult willbefound tobe Examples forsolution. (1)Find thecurve whose subnormal isconstant. (2)Thetangent atanypointPofacurve meets theaxis ofxinT. Find thecurve forwhichOP=PT, being theorigin. (3)Find thecurve forwhich theangle between thetangent and radius vector atanypointistwice thevectorialangle. (4)Find thecurve forwhich theprojectionoftheordinate onthe normal isconstant. Find theorthogonal trajectoriesofthefollowing families ofcurves : (5)sa-t/a=a*.(6)s*4-y*=a*. (7)px*+qy2**a*, (pandqconstant). aft (8)rfl-a. (9)r-^. (10)Find thefamilyofcurves thatcutafamilyofconcentric circles ataconstantangle a. 22 DIFFERENTIAL EQUATIONS MISCELLANEOUS EXAMPLES ONCHAPTER II. (1)(3y-*)|!-y.(2)xd^= ^(3)tanxcosydy-fsinydx+e*111*efcc=0. (4)o^+3t/2=zya .[Sheffield.] (5)a (6)show thatdx hx+by+f representsafamily ofconies. (7)Show that ydx-2xdy= representsasystemofparabolaswithacommon axisandtangentat thevertex. (8)Show that (4#+3y+1)dx+(3x+2y+1)dy=0 representsafamily ofhyperbolas havingasasymptotes thelines x4-y=and (9)If ~+2yta,idx andywhen x=%7r, show thatthemaximum value ofyis^. [Math. Tripos.] (10)Show that thesolution ofthegeneral homogeneous equation ofthe firstorderanddegree~=/ (- )is logs= J-x/ dv /(*)-*' where vy/x. (11)Prove thatxh y*isanintegratingfactor of pydx+qxdy+xmyn (rydx+sxdy)=*0 A+w+1 k+n+l .. ,If _ Qriri1^ ^a-cillv* y q Usethismethod tosolve 3ydx-2xdy +x^f-1(Wydx-6a?dy)-0. (12)Bydifferentiatingtheequation "*"8tx)+W(x)d(x) x jf(xy)-F(xy) xy verify that7 MISCELLANEOUS EXAMPLES 23 isanintegratfe^factor of Hence solve(x*yz-fxy+1)ydx-(xzyz- (13)Prove that iftheequationMdx+Ndy^Qisexact SN^dM dx By" [Foraproofoftheconverse seeAppendixA.] (14)Verify thatthecondition foranexactequationissatisfiedbj Hence show thatanintegratingfactor canalways befound for isafunction ofxonly. Solvebythismethod (15)Find thecurve(i)whosepolar subtangentisconstant; (ii)whose polar subnormal isconstant. (16)Find thecurve whichpasses throughtheorigin and issuch that thearea included between thecurve, theordinate, andtheaxis ofxisktimes thecube ofthat ordinate. (17)Thenormal PGtoacurve meets theaxis ofxinG. Ifthe distance ofGfrom theoriginistwice theabscissa ofP,provethatthe curve isarectangular hyperbola. (18)Find thecurve which issuch thattheportionoftheaxis ofx cutoffbetween theorigin andthetangentatanypointisproportional totheordinate ofthatpoint. (19)Find theorthogonal trajectoriesofthefollowingfamilies of curves: (i)(-l) +y+2B-0, (ii)r-a0, (iii)r=a+cosn6, andinterpretthefirst resultgeometrically. (20)Obtain thedifferentialequationofthesystemofconfocal conica ' andhence show thatthesystemisitsownorthogonal trajectory. (21)Find thefamilyofcurvescuttingthefamilyofparabolas 24 DIFFERENTIAL EQUATIONS (22)Ifu+iv^ffa +iy),where u,v,xandyare allreal,provethat thefamilies w==constant, v=>constant areorthogonal trajectories. AI ^x 92"32w .92vd*vAlsoprove that^2+5-*"*~a~i+3-1-ax2 tf!/2eto2dy2 [This theorem isofgreat useinobtaininglines offorceand lines of constantpotentialinElectrostatics orstream lines inHydrodynamics. uandvarecalled Conjugate Functions.] (23)The rate ofdecay ofradium isproportionaltotheamount remaining. Prove thattheamount atanytime tisgivenby A-A*-*. (24) If-rs*9\l~p)andv=0 if*0, provethat v*&tanh. [This gives thevelocityofafalling bodyinair,cakingtheresistance oftheairasproportionaltov2 .As tincreases, vapproachesthelimiting value k.Asimilar equation givestheionisation ofagasafter being subjectedtoanionising influence fortimet.] (25)Twoliquidsareboilinginavessel. Itisfound thattheratio ofthequantitiesofeachpassingoffasvapouratanyinstant ispro- portional totheratio ofthequantitiesstillintheliquidstate. Prove that thesequantities (sayxandy)areconnected byarelation ofthe form'y-cx*. [From Partington's Higher MathematicsforStudentsof p.220.] CHAPTER III LINEAR EQUATIONS WITH CONSTANT COEFFICIENTS 23.Theequationstobediscussed inthischapterareoftheform dn~!y dy wheref(x)isafunction ofx,butthep'sare allconstant. Theseequationsaremost importantinthestudyofvibrations ofallkinds, mechanical, acoustical, and electrical. This willbe illustrated bythemiscellaneous examplesattheendofthechapter. Themethods tobegiven below arechiefly duetoEuler and D'Alembert.* Weshall alsodiscuss systemsofsimultaneousequationsofthis form,andequationsreducible tothisformbyasimple transformation. 24.The simplest case;equations ofthe first order. Ifwetake n=land/(z)=0, equation (1)becomes (2) i.e.p 7 orPQlogy+px=constant, BOlogy=-Pix/p4-constant --;PizM>+log^> say, giving y**Ae~P&I**. 26.Equations ofthesecond order. Ifwetaken=2andf(x)=0, equation (1)becomes Jean-le-Bond D'Alembert ofParis (1717-1783)isbestknown by"D'Alem- bert's Principle"inDynamics. Theapplicationofthisprincipletothemotion offluids ledhimtopartial differential equations. 25 26 DIFFERENTIAL EQUATIONS Thesolution ofequation (2)suggeststhaty-Aemx ,wheremia some constant, may satisfy (3). With thisvalue ofy,equation (3)reduces to Aemx(pQm2+pm+p2)=0. Thus, ifmisaroot of flX+jyw+ft-O,...........................(4) y=<4ema>isasolution ofequation (3),whatever thevalue ofA. Lettheroots ofequation (4)beaand/3.Then,ifaand/3are unequal, wehavetwosolutions ofequation (3),namely y^Ae** andy^Be^. Now,ifwesubstitute y=AeaX+Beftxinequation (3),weshallget Ae**(p Qa*+p^a+p2)+Betx (pP*+Pl^+p2)=0, which isobviously true asaand^8aretheroots ofequation (4). Thus thesumoftwosolutionsgivesathird solution(thismight havebeen seen atoncefrom thefactthatequation (3)waslinear). Asthisthird solution contains twoarbitrary constants, equalin number totheorder oftheequation, weshallregarditasthegeneral solution. Equation (4)isknown asthe" auxiliary equation." Example. Tosolve2y-+5+2y=0 puty=*Aemxasatrial solution. Thisdx2dxy^y leads to Aemx(2m*+5m+2)=0, which issatisfied bym=-2or-. Thegeneralsolution istherefore 26.Modification when theauxiliary equation hasimaginary or complex roots. When theauxiliary equation (4)hasroots ofthe formp+iq, p- iq,where i2=-1,itisbesttomodify thesolution y-Aett+^+Bete-W*,...........................(5) soastopresentitwithoutimaginary quantities. Todothisweusethetheorems(giveninanybookonAnalytical Trigonometry) ew=cosqx+isinqx, e~^x=cosqx-isinqx. Equation (5)becomes y=epx(A(cosqx+isinqx)+B(cosqx-isinqx)} =epx{Ecosqx+Fsinqx}, writingEforA+BandFfori(A-B).EandFarearbitrary LINEAR EQUATIONS WITH CONSTANT COEFFICIENTS 27 constants, justasAandBare. Itlooks atfirstsightasifFmust beimaginary, butthis isnotnecessarilyso.Thus,if iandB=l-2i, -2andJF--4. leads totheauxiliary equation whose roots arem32t. Thesolution maybewritten as orinthepreferable form y=e3aj ($cos2z+J1sin2x), oragainasy=Ce3xcos(2x-a), where Ccosa=-Eand sina=-F, sothat C~J(E* +F2 )andtana-F/E. 27.Peculiarity ofthecase ofequal roots. When theauxiliary equationhasequalroots a=/3,thesolution reduces to y=(A+B)e*. NowA+J5,thesumoftwoarbitrary constants,isreally onlya single arbitrary constant. Thus thesolution cannot beregardedas themostgeneralone. Weshallprovelater(Art. 34)thatthegeneralsolution is 28.Extension toorders higher than thesecond. Themethods ofArts. 25and26applytoequation (1)whatever thevalue ofw,as longas/()=0. Theauxiliary equationis givingm=l,2,or3. Thus y-Ae*+Be2*4-Ce3*. Theauxiliary equation wm8-8=0, i.e.(m-2) (w*+2m+4)-0, giving m=2or-lt\/3. Thus y-Ae2*+e~*(E cosx^/3+Fsin or y=>4e2*4-Ce""* cos(x\/3-a). 28 DIFFERENTIAL EQUATIONS Examples forsolution. Solve ,6, +4 , +!.. x' 3 a (8)What doesthesolution tothelastexample become iftheinitial conditions are fa y-l, ~-**Q when x0, and ifyistoremain finitewhen x=+oo? Solve (11) +8y=0. (12) -64y- (13)l-jx+gQ-O, giventhat0-aand--=0 when <=0. [Theapproximate equationforsmall oscillations ofasimple pen- dulum oflength I,startingfrom rest inapositioninclined atatothe vertical.] (14)Find thecondition thattrigonometrical terms shouldappear inthesolution offl*s [The equationofmotion ofaparticleofmass m,attracted toa fixedpointinitslineofmotion byaforce ofctimes itsdistance from thatpoint, anddamped byafrictional resistance ofktimes itsvelocity. Theconditionrequired expressesthatthemotion should beoscillatory, e.g.atuning forkvibratinginairwhere the elastic force tendingto restore ittotheequilibrium positionisproportionaltothedisplacement andtheresistance oftheair isproportionaltothevelocity.] (15)Prove that ifkissosmall thatk2/mcisnegligible,thesolution oftheequationofEx.(14)isapproximatelye~kt^mtimeswhat itwould beifkwere zero. [This shows thatslight damping leaves thefrequency practically unaltered, butcauses theamplitudeofsuccessive vibrations todiminish inageometric progression.] JLINEAR EQUATIONS WITHCONSTANT COEFFICIENTS 29 (16)SolveLj|+^+^=0,given thatQ~Q Qand^=0when f0,andthatCR2<L. [Qisthechargeattime tononeofthecoatingsofaLeyden jarof capacity C,whosecoatingsareconnected when 2=byawire ofresist- anceRandcoefficient ofself-induction L.] 29.TheComplementary Function andtheParticular Integral. So farwehave dealt onlywithexamples where the/(#) ofequation (1) hasbeenequaltozero.Weshallnowshow therelation between thesolution oftheequation when f(x)isnotzeroandthesolution ofthesimpler equation derived from itbyreplacing f(x)byzero. Tostart withasimple example,consider theequation Itisobvious thaty=xisonesolution. Such asolution, con- taining noarbitrary constants,iscalled aParticularIntegral. Now ifwewritey~x+v, thedifferentialequationbecomes .e.a+B ax* ax givingv=Ae~*x sothat y=z+Ae~2x Theterms containing thearbitraryconstants arecalled the ComplementaryFunction. Thiscaneasilybegeneralised. Ify=uisaparticular integralof lf , dnu dn~lu du f../fr. BOthatj>--+Pl i-f...+JV* +pnu=/(x),.........(7) puty*=*u+vinequation (6)andsubtract equation (7). Thisgives dnv dn"1v dv ~/ov Ifthesolution of(8)bev=F(x), containing narbitrary con- stants, thegeneralsolution of(6)is andF(x)iscalled theComplementaryFunction. P.D.H. D 30 DIFFERENTIAL EQUATIONS Thus thegeneralsolutionofalineardifferential equationwith constantcoefficientsisthesumofaParticularIntegral and theCom- plementary Function, the latterbeingthesolutionoftheequation obtained bysubstitutingzeroforthefunction ofxoccurring. Examples forsolution. Verify thatthegiven functions areparticular integralsofthefollow- ingequations, and findthegeneral solutions : n\e*.^_o^4-2w-e* M1^n W '<&2dxy~~'W*'5^~1'3 d*u (3)2sin3z; ^4+4?/--10 sin3x. Forwhat values oftheconstants arethegiven functionsparticular integralsofthefollowing equations? (5)aebt ;~-f9s=GOe'1 .(6)asinpx ;-=-|-I-y=12sin 2as. (7)asinpx+bcospx ;y^+4-~+3y=8cos cc-6sinx. ii'jc dx (8)a; +B5+6sr-- Obtain, bytrial, particular integralsofthefollowing: (11) +9</=40sin5z.(12) -8+97/=40sin 30.Theoperator Dandthefundamental laws ofalgebra. When aparticular integralisnotobvious byinspection,itisconvenient toemploycertain methodsinvolvingtheoperator Z),which stands for -7-. Thisoperatorisalsouseful inestablishing theform ofthe complementary function when theauxiliary equationhasequal roots. d2d3D2willbeused for^r^,D3for^-^,andsoon.dx2dx* LINEAR EQUATIONS WITH CONSTANT COEFFICIENTS 31 Tbeexpression2j^-f5-+2ymaythen bewritten or Weshalleven write thisinthefactorised form faetorising theexpressioninDasifitwereanordinary algebraic quantity.Isthisjustifiable? Theoperations performedinordinary algebraarebased upon three laws : I.TheDistributive Law II.TheCommutative Law a6=6a; III.TheIndexLawam .an=am+fl . NowDsatisfies the firstandthird ofthese laws, for and Dm.Dnw=Dm+n.M (mandnpositive integers). Asforthesecond law,D(cu)=*c (Du)istrue ifcisaconstant, butnot ifcisavariable. Also Dm(Dnu)-Dn(Dmu) (mandnpositive integers). ThusDsatisfies thefundamental laws ofalgebra exceptinthat itisnotcommutative with variables. Inwhat follows weshall writeF(D)=pDn+p1Dn~l+...+pn^D+pn, where thep'sareconstants andnisapositive integer. Weare justifiedinfaetorisingthis orperforming anyotheroperations depending onthefundamental laws ofalgebra. Foranexample ofhow thecommutative law foroperatorsceases toholdwhen negative powersofDoccur, seeEx.(iii)ofArt. 37. 31.F(D)eax=eaxF(a). Since andsoon, F(D)e*= (Po +.-+2W +Pn)<?* 32 DIFFERENTIAL EQUATIONS 32.F(D){eaxV}=eaxF(D -fa)V,whereVisanyfunction ofx.By Leibniz's theorem forthewthdifferential coefficient ofaproduct, (Dn e**)V+n(Dn~1eax)(DV) n(n-l)an~2eaxD*V+...+eaxDnV ...+Z)")F Similarly D*-l{**V}=eP*(D +a)*-lV,andsoon. Therefore -*+...+pn_J)+pn){e*xV} l+...+pn-i(D+a)+pn}V 83.F(D2 )cosax=F(-a2 )cosax. Since D2cosax=-a2cosax, D4cosax= (-a2 )2cosax, andsoon, F(D*)cosoa;=(pD2n+p1D2"-2+...+pn_1Da+yn)cosax -(Po(-a )n+ft(-a2 )11-14-...+y j|^(-a2 )+pn}cos =J(-a2)cosao:. Similarly F(D2 )sinaz=F(-a2 )sinax. 34.Complementary Function when theauxiliary equation hasequal roots. When theauxiliary equation hasequalroots aand a,it maybewritten ma_%ma+aa=Q. Theoriginaldifferentialequationwillthenbe (Z)-a)2y=0 ............................(9) Wehavealready found thaty=Aerxisonesolution. Tofind amoregeneral oneputy=e*xV,whereFisafunction ofx. ByArt. 32, Thusequation (9)becomes D*F=0, i.e.V*=A+Bx, sothat LINEAR EQUATIONS WITH CONSTANT COEFFICIENTS 83 Similarlytheequation (D- a)*>//= reduces to DpV=0, giving V=(Al4-A2x+A^tf+ ...+Apx*~l ), andy=e(LX(Al+^42+A3x*+ ...+^z*-1 ). When there areseveralrepeated roots, asin (D~ar(D-p)<i(D-yyy=0,.....................(10) wenote that astheoperators arecommutative wemay rewrite the equationintheform which istherefore satisfied byanysolution ofthesimpler equation (D-a)*y-0 ...............................(11) Similarly equation (10)issatisfied byanysolution of (D-/3)^=0, ..............................(12) orof(D-y)ry=0 ...............................(13) Thegeneral solution of(10)isthesum ofthegeneral solutions of(11), (12),and(13), containing together (p+q+r)arbitrary constants. Ex.(i).Solve (#4-8D2+16)t/=0, i.e.(Z>2~4)2 ?/=0. Theauxiliary equationis(w2~4)2=0, w=2(twice) or-2(twice). Thusbytherulethesolution is y(A+Bx)ezx+(E+Fx)e~2*. Ex.(ii).Solve (Z)2-fl)2 t/=0. Theauxiliary equationis(m2-fI)20, m=i(twice) or- 1*(twice). Thus y-(A+Bx)e**+(E+Fx) e'**, orbetter y(P-fQx)cos a;+(R+6'x)sin a?. Examples forsolution. (1)(D4+2D3+Z>2)t/=:0. (2)(D6+3D4-f3Z)2-fl)t/-0. (3)(Z)4-2/)3+2Z)2~2Z)+l)y-0. (4)(4D5-3Z)3-D2 )y=0. (5)Show that F(D2 )(Pcoshax-fQsinhax)=F(a2 )(Pcoshaz-fQsinhaz). (6)Show that(D-a)4n (eaxsinpx)=*p*neaxsin^pa:. 35.Symbolical methods offinding theParticular Integral when f(x)^e8-*.Thefollowing methods areadevelopmentoftheidea oftreating theoperatorDasifitwereanordinary algebraic quan- 34 DIFFERENTIAL EQUATIONS tity.Weshallproceed tentatively,atfirstperforming anyopera- tions thatseemplausible, andthen,when aresult hasbeen obtained inthismanner, verifyingitbydirect differentiation. Weshall use thenotation-p(^f(x)todenote aparticular integraloftheequation F(D)y-f(x). (i)Iff(x)=eax ,theresult ofArt. 31, suggests that, aslongasF(a)=J=Q,-^n~\e*xmay^eavaluef#v7T\e<1*' Thissuggestioniseasily verified, for (ii)IfF(a)=0,(D-a)must beafactor ofF(D). Suppose thatF(D)~(D-a)<f>(D), where Then theresult ofArt. 32, euggeststhatthefollowing maybetrue, ifVis1, U^LLI * __ =__-- F(D) (D-a)P4>(D) (D-a)P\<f>(a) adoptingtheverynaturalsuggestionthat^istheoperatorinverse toD,that istheoperatorthatintegrates withrespecttox,while Y~pintegrates ptimes. Aga manner iseasily verified, for~ pintegrates ptimes. Againtheresult obtained inthistentative ap byArt. 32, byArt. 31. LINEAR EQUATIONS WITH CONSTANT COEFFICIENTS 35 Inworking numericalexamplesitwillnotbenecessarytorepeat theverification ofourtentative methods. Ex.(i). Theparticular integralis ~ (2+3)2 Addingthecomplementary function, weget Ex.(ii). (D Ifwesubstitute 2forDiny-~^50e2aj ,wegetinfinity.(D 2) Butusingtheother method, Adding thecomplementary function, weget y-26z2e2*+( Examples forsolution. Solve (1)(D*+6D+25)y=101<?*.(2) (3)(D2-9)?/-54e3a! .(4) (5)(D2-7)2)?/-a8inh^x. (6) 36.Particular Integral when f(x)=cos ax.From Art. 33, <f>(D2 )cosax=(p(-a2 )cosax. Thissuggeststhatwemayobtain theparticular integral by writing-a2forD2wherever itoccurs. Ex.(i). (D2+3D 4-2)y=cos2x. ^.cos2=-^=:- .cos2z=;r~s.cos2x.D2-f37)4-2"w"*"--4 +3D+2'wo *Ii-/3D-2 TogetD2inthedenominator, trytheeffect ofwriting ^ 3Z)-2""9Z)2-4> suggested bytheusualmethod ofdealingwith surds. ThisgivesOT\ iO cos2x-(3Z)cos2oj-f2cos^- .-T1 -(-6sin2x-f2 cos2x) 36 DIFFERENTIAL EQUATIONS Ex.(ii). (D3-f6D2-f-llZ)4-6)y2sin3aj. 1 -*o1 2sin3a=2on5/^^= ^a? 1 sinD-24 D+24 sin3^ Z>2-576 os3z+24sin3x) s3x+8sin3z). Wemaynowshow, bydirect differentiation, that theresults obtained arecorrect. Ifthismethod isappliedto [<f>(D2 )+Zty(D2 )]y-Pcosaz+Qsinas, where P,Qandaareconstants, weobtain (-a2 ).(Pcosor4-(?sina#)-fa\/r (-a2 ).(Psinax-Qccsax) Itisquite easy toshow that this isreallyaparticular integral, providedthatthedenominator doesnotvanish. Thisexceptionalcase istreated later (Art. 38). Examples forsolution. Solve (1)(D+l)y-10sin2aj. (2)(D*-5Z> +6)y-100sin 4a?. (3)(D2+8D+25)t/=>48cosa;--168mx. (4)(Dz-f2D-f401)y-sin20x+40cos20x. (5)Prove thattheparticular integralof d*srt,ds n dfi+ndt*P qt maybewritten intheform 6cos(qt- f), where 6-a/{(p*-g2 )2-f4F?2 }*and tanc-2lg/(p-?2 ). Hence prove that ifqisavariable and k,pandaconstants, 6is greatest whenq***\/(p2-2k2)=*papprox.ifA;isvery small, andthen c=7T/2approx and6a/%kp approx. [This differentialequationrefers toavibrating system damped byaforceproportionaltothevelocity andacteduponbyanexternal periodicforce. Theparticular integral givestheforced vibrations andthecomplementaryfunction thefreevibrations, which aresoon damped out(seeEx.15followingArt.28).Theforced vibrations have thegreatest amplitudeiftheperiod 2?r/joftheexternal force isvery nearly equaltothat ofthe free vibrations (whichiff LINEAR EQUATIONS WITH CONSTANT COEFFICIENTS 37 approx.), andthen ethedifference inphase between theexternal forceandtheresponseisapprox. -rr/2. This istheimportant phenomenonofResonance, which hasimportant applicationstoAcoustics, Engineering andWirelessTelegraphy.] 37.Particular integral whenf(x)=xm ,wheremisapositive integer. Inthiscasethetentative method istoexpand ynj^inaseries of ascending powersofD. Hence, adding thecomplementary function, thesolutionsuggested for is y=\(x*-J)+Acos 2>x+Bsin2x. Ex.(ii). bypartial fraction8j Addingthecomplementary function, thesolutionsuggestedfor Is **>-)- -9696'iVi2~T Hence thesolution ofD2 (Z)a+4)y=96x2should be y-2x*-6x*+4coa2x+Bsin2x+E+Fx. Alternative method. 38 DIFFERENTIAL EQUATIONS Thisgives anextra term 3,whichis,however, included inthe complementary function. *Themethod adoptedinExs.(i)and(ii),where F(D) does not containDasafactor, maybejustifiedasfollows. Supposetheexpan- sions have been obtained byordinary longdivision. This isalways possible, although theuseofpartialfractions maybemore convenient inpractice.Ifthedivision iscontinued until thequotientcontains Dm , theremainder willhaveDm+1asafactor. Call it<J>(D).Dm+1 .Then (1) This isanalgebraical identity, leadingto l=F(D){c Q+clD+c2D2+...+cmD}+<f>(D).D+1.......(2) Nowequation (2),which istruewhenDisanalgebraical quantity, isofthesimple formdepending onlyontheelementarylaws ofalgebra, which have beenshown toapplytotheoperator D,and itdoes noi involve thedifficulties which arisewhen division byfunctions ofDis concerned. Therefore equation (2)isalsotruewhen each side ofthe equationisregardedasanoperator. Operating onxmweget,since ............(3) which provesthat theexpansion obtained in(1),disregardingthe remainder, suppliesaparticular integralofF(D)y~xm . Itisinterestingtonote that thismethod holds good even ifthe expansion would bedivergentforalgebraicalvalues ofD. Toverifythe firstmethod incases likeEx.(iii),wehave toprove that 1 i.e. isaparticular integralof(F(D).Dr } i.e.that(F(D).jy}{(cJ) +...+cmD~r+m)xm }=xm................ (4) Now(F(D).Dr}u~F(D).{Dr u}, also^{(cgD~r+')xm }-(c8D)xm ; hence theexpression ontheleft-hand side of(4)becomes F(D){(c +ClD+ctD*+...+cmD>)x>}=x<,by(3), which iswhatwastobeproved. Inthealternative method wegetrextra terms intheparticular integral, say(^D-H***... +WD")*. Thesegive termsinvolving the(r-l)fchandlower powersofx. Butthese alloccur inthecomplementaryfunction. Hence the first method ispreferable. *The restofthis article ihould beomitted onafirstreading. LINEAR EQUATIONS WITH CONSTANT COEFFICIENTS'39 Note that ifD~ludenotes thesimplest form oftheintegralofu, without anyarbitrary constant, while BOthatD(D~l .l)=f=D~l.(7).1V Similarly Dm(D~m .xn)=D-m(Dw .xn )tifmisgreater than n. Sowhennegative powersofDareconcerned, thelaws ofalgebra arenotalways obeyedThisexplains whythetwo different methods adoptedinEx.(iii)givedifferent results. Examples forsolution. Solve(1)(D+l)y=x*.(2)(D2+2Z))//=24z. (3)(D2-6D+9)i/=54x+18.(4)(D*- 6Z>3+9D%=54z-f 18. (5)(D2-D-2)y=44-76x-48z2 . (6)(D3-D2-21%-44-76z-48x2 . 33.Particular integrals inother simple cases.We shall now givesometypical examplesoftheevaluation ofparticular integrals insimplecaseswhich have notbeen dealt with inthepreceding articles. Thework istentative, asbefore. Forthesake ofbrevity, theverfication isomitted, asitisverysimilar totheverifications already given. Ex.(i). Wecannot evaluate- 2^sin2xbywriting-22forD2 ,asin Art. 36,forthisgiveszero inthedenominator. But tsin2xistheimaginary partofe2ix ,and 1 * .1, asinArt. 35, 1 2x+isin2a?) 40 DIFFERENTIAL EQUATIONS hence, pickingouttheimaginary part, -.sin2x=-1$cos 2a/. Adding thecomplementary function, weget y=Acos2z+Bsin2z-J#cos2x. Ex.(ii). (D2-5Df6)*/=e2xx*. * -e2x (-Jx4-a;3-3z2-6z-6). Adding thecomplementary function, weget yAer* e- includingtheterm-6e2xin Ex.(iii). (D2 .S<?*sin2x-8^Vn3\2_ 6X (n 3)131sin 5 1 8e3a! (- ^a;cos2x) (seeEx.(i)) Adding thecomplementary function, weget y=e?x(Acos2x+Bsin2x-2xcos2z). These methods aresufficient toevaluate nearlyalltheparticular integrals that thestudent islikelytomeet. Allother casesmay bedealt withonthelines indicated in(33)and(34)ofthemiscel- laneousexamplesattheendofthischapter. Examples forsolution. Solve (1)(D2+l)t/=4cosz.(2)(Z>-l)t/-(z +3)e2*. (3)(D3-3D-2)y540x8e-a '.(4)(Z)2+2D+2)y=26-* sinx. (5)(D2+])'t/=24zcosz.(6)(Z>5-D)t/=12e*+8sinx-2x. (7)(Z>2-6D+25)y=W*cos4z+8e3*(l-2z)sin4s. 39.TheHomogeneous Linear Equation. This isthenamegiven botheform(px*Dn+/^a;*-1/)*-14- ...-fpn)y=/ (z*). Itreduces tothetypeconsidered before ifweputx=e*. LINEAR EQUATIONS WITH CONSTANT COEFFICIENTS 41 Ex. (j?IP+Zx*D*+ Put x-tf, Ax . .othat Z>'*J";axdxatxat xdt rfd*\ 2/ dd*\ 1 /dd* thus thegiven differential equation reduces to-j~at giving t/-^4+Bt+Ct*+3e2' ~A+Blogx4-C(log a:)24-3z2 . Another method isindicated in(28)-(30)ofthemiscellaneous examplesattheendofthischapter. Theequation Po(+bx)nDny+p l(a+bx)"-1!)"-1y+...+p ny-f (x) canbereduced tothehomogeneouslinear form byputting -a+fe,gmng -=dyJydz =f)dy ,ydxdzdx dz Examples forsolution. (3) (5) (6)(l+*)a 42 DIFFERENTIAL EQUATIONS 40.Simultaneous linear equations with constant coefficients. The method willbeillustrated byanexample. Wehave two de- pendent variables, yandz,andoneindependentvariable x. Dstands for-y-,asbefore.ax Consider (5D+)y-(2D +l)z~e-*,.....................(1) (JD+8)y-3z =56-' ......................(2) Eliminatez,asinsimultaneous linearequationsofelementary algebra. Todothiswemultiply equation (1)by3andoperateon equation (2)by(2D+1). Subtractingtheresults, weget {3(5Z>+4)-(2D+1)(D+8)}y-3e~*-(2D+1)56-*, i.e.(~2D*-2D+)y=86-, or (D2+D-2)y=- Solvingthis intheusualway,weget The easiest way togetzinthisparticular exampleistouse equation (2),which doesnotinvolve anydifferential coefficients ofz. Substitutingforyin(2),weget Ue~x+$Aex+$Be-*x-3z=5e~*, soth'at z=3e~x+3Aex+2Be~2x . However, when theequations donotpermitofsuch asimple method offinding z,wemayeliminatey.(But seep.48.) Inourcase thisgives {-CD-f8)(2Z> +l)+3(5Z> +4)}Z=(D+8)<r-(5D +4)5<r, i.e.(-2D2-21)+4)z-12<r, giving2=Ser*+J?e*+^e-2a! . Tofindtherelation between thefourconstants A,B, ,andF9 substitute ineither oftheoriginal equations, say (2).Thisgives (D+8)(26-*+4e*+<r2 *)-3(3e~x+Eex+Fe~2x}=5e~x 9 i.e.(94-3E)e*+(65-3F)e~*x-0, whence E=34 andF=2B, so zSe-35+#e* -fJPe-2e-3erx+S^e254-25e"2a! ,asbefore. Examples forsolution. (1)Dy-z -0, (2)(Z>-17)y+(2Z>- 8)z=0, (3)(2D2-D-f9)t/-(Z)2 +Z)-f3)z-0, LINEAR EQUATIONS WITH CONSTANT COEFFICIENTS 43 (4)(/>+l)y- +6a , (5) (6) y-(D-S)z~ 29e-* -f47sin2z+23cos 2a?. MISCELLANEOUS EXAMPLES ONCHAPTER III. Solve (1)(Z)-l)8 2/=16e3a! .(2) (3) (4) (5)(Z)4-6Z>2-8Z>-3)^== 256(3 (6)(Z)4-8Z)2-9)?/=50sinh2z.(7)(Z>4-2D2+1)y=40cosh (8)(D-2)2 !/=8(x2+e2a:4-sin2x). (9)(D-2)2 ?/=8x2e2a:sin 2aj. (11) (12)(D-a)ay=ax ,where aisapositive integer. (13)^i1*.'U28 ,,4)<i'2>_ 1'' ''' ' (ID)-.(16)(, <>-*- s-1"I'1"S-fc(W.'S+.-Oi|-a (21)Show that thesolution of(D2n+l-l)y=Qconsists ofAe*and npairsofterms oftheform e*(Brcossx-fOrsinsx), 2-Trr . .27rr where Css=cos2^+land 5s=Sm 2iHhT' rtaking thevalues 1,2,3...nsuccessively. (22)If (Z)-a)w=0, (D-a)v=t, and (D-a)i/=v, findsuccessively w,v,andy,andhence solve (Z)-a)8y0. 44 DIFFERENTIAL EQUATIONS (23)Show thatthesolution of ( canbewritten Ae?x Hence deduce thesolution of(D-a)3*/=0. [Thismethod isduetoD'Alembert. Theadvanced student will notice that itisnotquite satisfactory without further discussion. It isobvious thatthesecond differentialequationisthelimit ofthe first, but itisnotobvious thatthesolution ofthesecond isthelimit ofthe solution ofthefirst.] (24)If(D-a)V11*iadenoted by z,prove thatz,~~,and|^all vanish whenm- a.dm dm Henceprovethat eax ,xea* tandx2ea*areallsolutions of(D- a)82/=0. [Note that theoperators (Z)-a)3and^arecommutative.] /n*x cii xi^ cosax-cos(a-fh)x (25)Show that isasolution of (D2-fa2 )y=cos(a+A)a?. Hence deduce theParticularIntegralof(D2+a2 )f/=cosax. [ThisisopentothesameobjectionasExample 23.] (26)Prove that ifFisafunction ofxandF(D) has itsusual meaning, (i)Dn[xV] (ii)F(D)[xV] =x (iv)0(/>)[xnF]-xn0(J9)F4-wxw-1^^)^+.--+n^n~rr(^ , f/rix, ,, 1 +... to(n-fl) terms,where 0(^0) stands for (27)Obtain theParticularIntegralsof(i)(D-I)y=o:e2a: , (ii)(D-f !)?/==x2cosa, byusingtheresults(iii)and(iv)ofthelastexample. (28)Prove, byinduction orotherwise, that if6stands forx-=-f n dx (29)Prove that (i) "provided whoreFisafunction ofx, LINEAR EQUATIONS WITHCONSTANT COEFFICIENTS 45 (30)Byusingtheresults ofthelastquestion, provethatthesolu- tionof where aandbaretheroots ofm(m-T)-4m+6=0, i.e. 2and 3. (31 )Given that(D-1)y-e2 *, provethat (D-1)(D-2)y=0. Bywriting down thegeneral solution ofthesecond differential equation (involving twounknown constants) andsubstitutinginthe first, obtain thevalae ofoneofthese constants, henceobtaining the solution ofthefirstequation. (32)Solve-~^-f-p2 2/=sinaxbythemethod ofthelastquestion. (33)Itu2denotes ea* \ue~axdx9 M2denotes e*** \Ujer** dz, etc., prove thesolution ofF(D)y~u, where F(D)istheproductofn factors maybewritten y^^tr This istrueeven ifthefactors ofF(D) arenot alldifferent. Hence solve (D-d) (D-b)y=eaxlogx. (34)Byputting ,,^.intopartial fractions, prove thesolution of F(D)y~u maybeexpressedintheform ^-ea *Juer**dx, providedthefactors ofF(D) are alldifferent. [Ifthefactors ofF(D) arenot alldifferent, wegetrepeatedinte- grations.] Theoreticallythemethods ofthisexample andthelastenable usto solve anylinear equation with constant coefficients.Unfortunately, unless uisoneofthesimplefunctions(productsofexponentials,sines and cosines, andpolynomials)discussed inthetext,wearegenerally leftwithanindefinite integration which cannot beperformed. Ifu=/(#), wecanrewrite e*x 1ue~axdx intheform \J(t)*<*-'*dt,JL where thelower limit kisanarbitrary constant. F.D.X. V 48 DIFFERENTIAL EQUATIONS (35) (i)Verify that 1f* y^-]f(t)a\np(x-t)dt isaParticularIntegralof g+j>ww. [Remember that ifaand6arefunctions of#, (ii)Obtain thisParticularIntegral byusingtheresult ofthelast example. (iii)Hence solve (D2+l)t/=cosec x. (iv)Show that thismethod willalsogivethesolution of (inaform freefromsignsofintegration), itf(x)isanyoneofthefunc- tions tanx,cota,secx). (36)Show thattheParticularIntegralof~+p2y=kco8pt repre- cit' sentsanoscillation withanindefinitely increasing amplitude. JThisisthephenomenonofRESONANCE, which wehavementioned before(seeEx.5followingArt.36). Ofcourse thephysical equations ofthistypeareonlyapproximate,soitmust notbeassumed that the oscillationreally becomes infinite. Still itmay become toolarge forsafety.Itisforthisreason that soldiers breaksteponcrossinga bridge,incase theirsteps might beintunewith thenatural oscillation ofthestructure.] (37)Show thattheParticularIntegralof represents anoscillation withavariableamplitude ^-te"**. Find themaximum value ofthisamplitude, andshow that itisvery largeifhisverysmall. What isthevalue oftheamplitudeafteran infinite time ? [This representstheforced vibration ofasystem which isinreson- ancewith theforcing agency, when both aredamped byfriction. The result shows that ifthis friction issmall theforced vibrations soon becomelarge, thoughnotinfinite asinthelastexample. This isan advantageinsome cases. Ifthereceivinginstruments ofwireless telegraphy were notinresonance with theHertzian waves, theeffects would betoofaint tobedetected.] LINEAR EQUATIONS WITH CONSTANT COEFFICIENTS 47 d*u (38)Solve M-rc4y0. eta/ [This equation givesthelateraldisplacement yofanyportionofa thin vertical shaft inrapid rotation, xbeingtheverticalheightofthe portion considered.] (39) If,inthelastexample, (Lii-=^=whena?0 and x*=*l,&x provethat y=E(coa nx-coshnx)-f^(sin nx-sinhnx) and cosnlcosh nl=>1. [Thismeans thattheshaft issupportedattwopoints, oneaheight Iabove theother, and iscompelledtobevertical atthesepoints. The lastequation givesnwhen Iisknown.] (40)Prove thattheComplementary Function of becomesnegligible when tincreasessufficiently, while that of oscillates withindefinitely increasing amplitude. [Anequationofthistypeholds approximatelyfortheangular velocityofthegovernorofasteam turbine. The firstequationcorre- spondstoastable motion ofrevolution, thesecond tounstable motion or" hunting.99SeetheAppendixtoPerry's Steam Engine.} (41)Prove thatthegeneral solution ofthesimultaneousequations: d2y dxmdt-He di' where m,F,H,and eareconstants, is x=*A V TJ where o>=andA,B,Cyaarearbitraryconstants. Given that -=-=-^=o;=y='0when J=0,show that these reduce to Vx--^(1-coacot), (l)/l y y**=j(wt-sino><),theequationsofacycloid. 48 PIFFERENTIAL EQUATIONS [These equations givethepath ofacorpuscleofmassmandcharge erepelled from anegatively-chargedsheet ofzinc illuminated with ultra-violetlight, under amagneticfieldHparallel tothesurface. Vis the electricintensity due tothechargedsurface. Byfindingex- perimentally thegreatest value ofx,SirJ.J.Thomson determined nr,fromwhich theimportantratio iscalculated whenVandHare known. SeePhilMag.Vol. 48,p.547,1899.] (42)Given thesimultaneousequations, d*I2 whereL19L2,Mycvc2tEandpareconstants, prove that7tisofthe orm !cospt+Alcos(mt-a)-fBlcos(nt- /3), and7j|oftheform a2cospt+A2cos(mt-a)-fB2cos(nt- ), Ewherea^-^pc^l-p*c2L2), EM, 2=-^-^iC2, kdenoting theexpression (LtL2-M2 )c^zp*-(ZjCj+L2c2)p*+1; mandnarecertaindefinite constants"; AltBltaand/?arearbitrary constants;andA2isexpressibleinterms ofAlandB2interms ofBv Prove further thatmandnarereal ifL19L&M9cl9andcaarereal andpositive,andL^L^>M2 . [These equations givetheprimary andsecondary currents/jand 7tinatransformer when thecircuits contain condensers ofcapacities Cjand c2.LTandL2arethecoefiicients ofself-induction andMthat ofmutual induction. Theresistances (whichareusually very small) havebeen neglected. Esinptistheimpressed E.M.F. oftheprimary.] Alternative methods forsimultaneous equations. InEx. 3,p.42, having foundy,wecanfind zwithout integration byoperatingon thegiven equations byDand(D+2)respectively andsubtracting. Given /(D), F(D), anytwopolynomialsinDwithnocommon factor containing Dtwecanfindother polynomials </>(#), \[s(D)>suchthat (D)f(D)- >/r(D)F(D)-1.(Cf.Smith's Algebra,Art. 100.) Insimplecaseswecanobtain(f>(D), \fs(D) byinspection. Alternatively, wemay replace thegiven equationsofEx.3by theirsumand difference. Proceeding similarlyinEx. 4,wemay takev+zandy-zasnewvariables. CHAPTER IV SIMPLE PARTIAL DIFFERENTIAL EQUATIONS 41.Inthischapter weshall consider some ofthewaysinwhich partialdifferential equations arise, theconstruction ofsimple par- ticular solutions, andtheformation ofmorecomplexsolutions from infinite series oftheparticularsolutions. Weshall alsoexplainthe applicationofFourier's Series, bywhich wecanmake thesecomplex solutionssatisfy givenconditions. Theequationsconsidered include those thatoccur inproblems ontheconduction ofheat, thevibrations ofstrings,electrostatics andgravitation, telephones, electro-magnetic waves, and the diffusion ofsolvents. Themethods ofthischapterarechiefly duetoEuler, D'Alembert, andLagrange.* 42.Elimination ofarbitrary functions. InChapterI.weshowed how toformordinarydifferential equations bytheelimination of arbitraryconstants. Partial differentialequations canoften be formed bytheelimination ofarbitraryfunctions. Ex.(i).Eliminate thearbitrary functions /andFfrom y=f(x-at) +F(x+at).........................(1) Wegetg|=f(x-at)+F'(x+at) andd ^=f"(x-at)+F''(x+at).........................(2) Similarly-j---af'(x-at)+aF'(x+at) and --a*f"(x-at)+a*F"(x +at)...................(3) *Joseph Louis LagrangeofTurin (1736-1813), thegreatest mathematician of theeighteenth century, contributed largely toevery branch ofMathematics. He created theCalculus ofVariations andmuch ofthesubject ofPartial Differential Equations, andhegreatly developed Theoretical Mechanics and Infinitesimal Calculus. 49 50 DIFFERENTIAL EQUATIONS 3^y j_a2y dx^a* di*9 From(2)and(3),g-1, f*................................. (4) partialdifferentialequationofthesecond order.* Ex.(ii).Eliminate thearbitrary function /froma I--M!) , dz I.,/y\and =---/(-), etyxVEX 5s; 9^ _ so a;^-+?/3-=0. &e^<ty Examples forsolution. Eliminate thearbitrary functions from thefollowing equations: (1)z**f(x +ay). (2)z=f(x +iy)+F(x-iy), where i2=-1. (3)2;^/(o: cosa-f?/sina-a)4-F(#cosa-f?/sina-fat). (4)s=/(z2-?/2 ). (5)z- (6)z-tf 43.Elimination ofarbitrary constants. Wehave seen in ChapterI.how toeliminatearbitrary constants byordinary differentialequations.Thiscanalsobeeffected bypartials. Ex.(i).Eliminate Aandpfrom zAeptsinpx. Weget ~-2=-p2Aeptsinpx, and~~^=^Mepfsinpx ; therefore1=0. Ex.(ii).Eliminate a,6,and cfrom z=a(x-ft/)4-6(x-y)+abt-fc. Weget f?.8dx dz *This equation holds forthetransverse vibrations ofastretchedstring.Themostgeneral solution ofitisequation (I),whichrepresents twowavestravelling withspeed a,onetotherightandtheother tothe left. Seepp.61,218,256. PARTIAL DIFFERENTIAL EQUATIONS 61 But rru tTherefore Examples forsolution. Eliminate thearbitraryconstants from thefollowing equations: (1)2== Ae~p*cospx. (2)z=Ae~ptcosqxsinn/,where p2=q2+r1 . (3)2=ax+(l-a)t/ +6.(4)z=aa?+fy/+a2+62 . (5)z=(#-a)2+(*/-&)*. (6)az+b=*a*x +y. 44.Special difficulties ofpartial differential equations. Aswehave alreadystated inChapter I.,every ordinarydifferentialequation ofthenthordermayberegardedasderived from asolution con- taining narbitraryconstants* Itmightbesupposedthatevery partialdifferential equationofthenthorderwassimilarlyderivable from asolution containing narbitrary functions. However, this is nottrue. Ingeneralitisimpossibletoexpress theeliminant of narbitraryfunctions asapartialdifferential equationoforder n. Anequationofahigherorder isrequired, andtheresult isnot unique.f Inthischapter weshallcontent ourselves withfinding particular solutions. Bymeans ofthesewecansolve suchproblemsasmost commonlyarise fromphysical considerations.]; Wemayconsole ourselves forourinabilitytofindthemostgeneralsolutions bythe reflection that inthose caseswhentheyhavebeenfound itisoften extremelydifficult toapply them toanyparticular problem. *Itwillbeshown later (Chap. VI.) that incertainexceptionalcases an ordinarydifferential equation admits ofSingular Solutions inaddition tothe solution with arbitraryconstants. These Singular Solutions arenotderivable from theordinarysolution bygiving theconstantsparticular values, butareof quite adifferent form. tSee Edwards' Differential Calculus, Arts. 612and 513, orWilliamson's Differential Calculus, Art. 317. $Thephysicistwilltake itasobvious thatevery suchproblem hasasolution, andmoreover that this solution isunique. From thepointofview ofpure mathematics, itisamatter ofgreat difficultytoprove the first ofthese facts: thisproof hasonlybeen given quite recently bytheaidoftheTheoryofIntegral Equations (seeHeywoodandFr^chet'sUEquation deFredholm etsesapplications alaPhysique A[a/hdmatique). Thesecond fact iseasily proved bytheaidof Green's Theorem (seeCarslaw's ConductionofHeat, 2nded.p.14). Forexample, Whittaker hasprovedthatthemostgeneral solution of Laplace's equation U V~If(xcos t+ysin t+iz,t)dtt but ifwewish tofindasolutionriatisfyingcertain given conditions onAgiven surface, wegenerallyuseasolution intheform ofaninfinite eerieg. 52 DIFFERENTIAL EQUATIONS 45.Simple particular solutions. Ex.(i).Consider theequation ^~a""~j5~ (which gives thecon- duction ofheat inonedimension). Thisequationislinear. Now, in thetreatment ofordinarylinearequations wefoundexponentials very useful. This suggests z**e*+niasatrial solution.Substitutingin thedifferentialequation, weget which istrue if nw2af , Thus6+ isasolution. Changingthesignofw,C'mx+m<1^iisalsoasolution. Ex.(ii).Find asolution ofthesameequation that vanishes when *=+00 . Intheprevioussolutions toccurs inem*a**.This increases witht, sincew2a2ispositiveifmandaarereal. Tomake itdecrease, putm=ip,sothat ra2a2=-p2a2 . Thisgiveseipx-P2"**asasolution. Similarly e-tpx-rw* iaasolution. Hence, asthedifferentialequationislinear, e~p*a*t(Aeipx+Bei<fx )is abdution, which wereplace,asusual, by e-pW(E cospx+Fsinpx). 92zd2zEx.(iii).Find asolution of pi-j+pTa^Owhich shall vanish when y-+oo,andalsowhen x=0.Cy Putting a-gwa+wy, weget(m2+n8 )e"1**-^-0,sowa+n2=0. Thecondition when y=+oodemands thatnshould berealand negative, sayn-p. Then m^-.ip. Hence er^AeW* +Be-***)isasolution, t.c. e~py(Ecospx+Fsinp:r)isasolution. But s=>0 if3=0, so#=0. Thesolutionrequiredistherefore Fe~pysinpx. Examples forsolution. 92vd*v (1)3-^=^y,given thaty=0when x -fooandalsowhen <+oo. CrtZ/ vf (2)^-i,853-!;H^giventhat zisnever infinite(foranyrealvalues of c/a? CLoy xory),andthatz^Owhenx=0 ory=0. (3)=-+a=-=0,giventhat 2isnever infinite, andthat=- when PABTIAL DIFFERENTIAL EQUATIONS 53 327327327 (4)-W-,+-^-r+-~-r=0,given thatV when &+oo,when Ctt/<71/^ OZ* y-oo,andalsowhen 2=0. 92F92F (5)"a"!^ ;f~~;f*ventnat^*9never infinite, andthatV^C and ^7973F .,M()whensc^v^z^O.3z 3y53* 32792737 (6)-5-?+ ~-s=~v ,given thatF=0when =+00,when#=0 or <7X4 Cty4Ot I,andwhen */=0 or I. 46.More complicated initial andboundary conditions.* InEx.(lii) ofArt. 45,wefound Fer*w sinpxasasolution of d*z d*z Bx*dy*' satisfyingtheconditions that2=0 ify*>-fccorifa:==0. Supposethatweimposetwoextra conditions, fsayz=0 if#=J and z=lx~x* ifyforallvalues ofxbetween andL The firstconditiongivessinpZ0, i.e.pi=mr,where nisanyinteger. Forsimplicity wewillatfirsttake Iir>giving p=>n, anyinteger. Thesecond conditiongivesFsinpxtrx-x2forallvalues ofx between and ?r.This isimpossible. However, instead ofthesolutionconsistingofasingle term,we maytake Ftf-ysinx+Ftf-^sin2x-fFze~*ysin3x+ ..., since theequationislinear(ifthis isnotclear,cf.Chap.TIT.Art.25), giving pthevalues1,2,3,...andaddingtheresults. Byputting y=0andequatingtoTTX-x2weget FIsinx-fF2sin2x-f.F3sin3x4-... TTX-x*forallvalues ofxbetween and TT. Thestudent willpossiblythink thisequationasimpossibleto satisfyastheother, but itisaremarkable factthatwecanchoose values oftheF'athatmake thistrue. This isaparticularcase ofamoregeneral theorem, which we nowenunciate. *As tusually denotes timeandxandyrectangular coordinates, acondition uchas=0wheni=0 5acalled aninitial condition, while onesuch asz=0 if x=0, orifx=J,orify=a;>iscalled aboundary condition. tThis istheproblemoffinding thesteadydistribution oftemperatureina emi-infinite rectangular stripofmetal ofbreadth/,when theinfinite sides are keptatandthebaseat(Ix-x2 )*. 64 DIFFERENTIAL EQUATIONS 47.Fourier's Half-Range Series. Every function ofxwhich satisfies certain conditions canbeexpandedinaconvergentseries oftheform f(x)=!sinx+a2sin2x+aasin3z+...toinf. for allvalues ofxbetween and ?r(butnotnecessarilyforthe extreme values x=0andx=TT). This iscalled Fourier's*half-rangesine series. The conditions alluded toare satisfied inpractically every physical problem. f Similarly,under thesameconditions/ (x)maybeexpandedin ahalfrangecosine series 6+&!cos a?+62cos2x+63cos3x+ ...toinf. These arecalledhalf-rangeseries asagainstthe series valid between and2-Tr,which contains both sineandcosine terms. Theproofsofthese theorems areverylongand difficult.JHow- ever,ifitbeassumed that theseexpansionsarepossible,itiseasyto findthevalues ofthecoefficients. Multiplythesine series bysinnx,andintegrate termbyterm, giving If(x)sinnxdx=*0i\sinxsinnxdx+aasin2xsinnxdx+... . Jo Jo Jo Theterm withanasafactor is T sin2nxdx -1nTJJoan\sir Jo -cosnxxx- nsin *Jean Baptiate Joseph Fourier ofAuxerre (1768-1830) isbestknown asthe author ofLaTheorie analytique delachaleur. Hisseries arose inthesolution of problems ontheconduction ofheat. tItissufficientforf(x) tobesingle- valued, finite, andcontinuous, andhave onlyalimited number ofmaxima andminima between #=andx ir.However, these conditions arenotnecessary. Thenecessary andsufficient setofconditions hasnotyetbeen discovered. %Forafulldiscussion ofFourier's Series, seeCarslaw's Fourier's Series and Integrals andHobson's Theory ofFunctions. Theassumption that this ialegitimateisanotherpointthatrequires justification. PARTIAL DIFFERENTIAL EQUATIONS 55 Theterminvolving anyother coefficient, say a,.,is sinrxsinnxdx a,Isi Jo dCn r {*Joco?(n-r)x-cos(n-f r)x}dx a,rsin(n-r)xsin *" 2Lw-r~~n+r Jo~* Soalltheterms ontheright vanishexcept one. f*Thus I/(x)sinnxdx=%a n-w9 Jo 2f71^ or an=/(x)sinwxdx. Similarly,itiseasytoprovethat if f(x)=60+6!cosx4-62cos2x -I-... forvalues ofxbetween andTT,then and bn=/(x)cosnxdx forvalues ofnother than 0. 48.Examples ofFourier's Series. (i)Expand vrx-x2inahalf-rangesine series, valid between x=0 undX=TT. Itisbetter nottoquotetheformula established inthelast article. Let TTXx2=djsinx-f ct2sin2x+ct3sin3x-f ... . Multiply bysinnxandintegrate from toTT,giving (TTX-x2 )sinnxdx=anIsin2nxdx=-an,asbefore. Jo^ Now, integrating byparts, 1(TTX-x2 )sinnxdx=(TTX-xa )cosnx -f- I(TT-2x)cosnxdx Jo L JowJo [1T 2f7" -^(TT-2x)sinnx -H-^Isinnxdxn2" Jow2 J 2rT4"0 5cosnx=-= ifnisoddor ifniseven.n3L Jo"3 gThus on=3ifwisoddor ifniseven, giving finally 2_8 /' v 66 DIFFERENTIAL EQUATIONS (ii)Expand f(x)inahalf-rangeseries validfrom#=tox=TT,where f(x)mx between#0andx^ 2t 7Tand /(z)=w(7r-z) between x=~andz=7r. <6 Inthis casef(x)isgiven bydifferent analytical expressionsin different partsoftherange.* Theonlynoveltyliesintheevaluation oftheintegrals. Inthiscase If(x)sinnxdx=* If(x)sinnxdx+ If(x)sinnxdx Jo Jo J pj fir=Iwxsin ftdx -f- Im(7r a;)sinn#eta. Jo J$ Weleave therestofthework tothestudent. Theresult is 4wi -(sinx-\sin3x-f-ssin5x-^Bin7z4-...). Thestudent should draw thegraphofthegiven function, and compareitwith thegraphofthe firsttermandofthesum ofthe first twoterms ofthisexpansion. f 'Examples forsolution. Expand thefollowing functions inhalf-rangesine series, valid between x**andXTT: (1)1. (2)x.(3)x*.(4)cosz.(6)e*. (^)/(;r)=!from x tox=T,andfrom a?=-7-toTT,4 4 /(x)---(4x-7r)(37r-4x) from x=~to&= . (7)Which ofthese expansions holdgood (a)forx= ? (6)forxTT? 49.Application ofFourier's series tosatisfy boundary conditions. Wecannowcomplete thesolution oftheproblemofArt. 46. Wefound inArt.46that Ff-y sinx-fFze~*sin2x+F^vsinSo;-f... satisfied alltheconditions,if Flsinx+Fzsin2x+F3sin3x+...=7rx~a5t forallvalues ofxbetween and TT. *Fourier's theoremapplies even iff(x]isgiven byagraphwithnoanalytical expressionatall, iftheconditionsgiveninthefootnote toArt.47aresatisfied. Forafunctiongiven graphically,theseintegralsaredetermined byarith- metical approximation orbyaninstrument known asaHarmonic Analvser. fSeveral ofthegraphswillbefound inCarslaw's Fourier's Series andIntegrals, 2nd ed.,Chap.VII. More elaborate onesaregiveninthePhil.Mag., Vol.45(1898), PARTIAL DIFFERENTIAL EQUATIONS 67 InEx.(i)ofArt.48wefound that,between and TT, o (sina;-H^Vsin3x-fTfasin5x+...)=TTX-#a . Thus thesolutionrequiredis o snx+^e~ysm%x+ilT*~5ysmx+) 7T 50.Inthecasewhen theboundarycondition involved Iinstead ofTT,wefound Fe~pysinpxasasolution ofthedifferentialequation, andtheconditions showed thatp,instead ofbeingapositive integer n,must beoftheformn-n-fL Thus F^e-^1sinTrx/l+F^^nsin2/I+ ... satisfies alltheconditions if FlsinTrx/l+F2sm27rx/l+...**lx-x* forallvalues ofxbetween and I. 72 71 Put TTX/I^Z. Then fa-a;2-,(^-f ).TheFsarethus, 7T27T* times asmuch asbefore. Thesolution istherefore 072~ 3in-rrx/l+^e^^11sinSTTX/^+Ti3e-5^sinSTTCC//+...) MISCELLANEOUS EXAMPLES ONCHAPTER IV. 1 J^. (1)VerifythatV**r.e*Ktiaasolution of V* to(2)Eliminate ^4andpfrom7=^-^ sin dV (3)Transform -g--JT aw^ byputtingF-e-*'TF. [Thefirstequation givesthetemperatureofaconductingrodwhose surface isallowed toradiate heat into airattemperaturezero. The giventransformation reduces theproblemtoonewithout radiation.] (4)Transform 3VKd(a9F\.dWvd*W Tr*dr\rTr)*-W=K~W byputtingW=rV. [ThefirstEquation givesthetemperature of ftsphere,when heat flows radially.] 58 DIFFERENTIAL EQUATIONS (5)Eliminate thearbitrary functions from (6) (i)Show that ifemx+intisasolution of 3V 3*V-=K-^-hV, where nandharereal,thenmmust becomplex. (ii)Hence, putting m=>-g-if, show thatVe~vxsin(nt-foe)isa solution that reduces toFsinn< forx=0,provided K(g2-f2)=*hand (iii)If7=0when $=-+oo,show that ifKandnarepositive so aregand/. [InAngstrom's method ofmeasuring K(the" diffusivity "),one endofavery longbar issubjected toaperiodic change oftemperatureFsinnt.Thiscauses heatwaves totravelalongthebar.Bymeasur- ingtheirvelocity and rate ofdecay n/fandgarefound.Kisthen calculated fromK=*n/2fg.] dV (7)Find asolution of -=K^-reducingtoFsinnt forx= andtozero forx=+oo .dt x [Thisistheproblemofthelastquestion when noradiation takes place. Thebarmay bereplaced byasemi-infinite solidbounded by aplane face, iftheflow isalways perpendicular tothat face. Kelvin foundKfortheearth bythismethod.] (8)Prove thatthesimultaneousequations aresatisfied by V Z if g*-f*=RK-n*LC, and 72 (R+iLn)-V<?(K+iCn). [These areHeaviside^s equationsforatelephone cable with resist- ance R,capacity 0,inductance Ltandleakance K,allmeasuredper unitlength.7iathecurrent andVtheelectromotive force.] (9)Show that inthelastquestion gisindependentofnifRC=KL. [The attenuation ofthewave depends upon g,which ingeneral depends uponn.Thus,ifasound iscomposedofharmonic waves of differentfrequencies, these waves aretransmitted with different degrees ofattenuation. Thesound received attheother end istherefore MISCELLANEOUS EXAMPLES 59 distorted. Heaviside's device ofincreasing LandKtomakeRC=KL preventsthisdistortion.] (10)Inquestion (8),ifZ=*/T=0, show that bothVand /are propagated withvelocity \/(2n/RC). [The velocityisgivenbyn/f.] (11)Show thatthesimultaneousequations *?*L?Z-?- __!^3a^dR_dQt cdtdydz' cdtdydz' *?? ==^_?y. _^^ =?.?_<^?. cdt dzdx'cdtdzdx kdR =3d/3_dat_/5dy^?? _??. cdtdxdy'cdtdxdy' aresatisfied byP= ; a= ; Q=0; /3=/3sinp(x-vt); R=RQginp(x-vt) ;y=0; providedthatv^c/Vk/u. and/3=-VW/*) ^o- [These areMaxwell'selectromagnetic equationsforadielectric of specificinductive capacity kandpermeability //.P,Q,Rarethe componentsoftheelectricintensity and a,/3,ythose ofthemagnetic intensity,cistheratio oftheelectromagnetictotheelectrostatic units (whichisequaltothevelocityoflightinfreeether). Thesolution shows thatplane electromagnetic waves travel withthevelocity c/Vkjm, andthat theelectric andmagneticintensities areperpendicular tothe direction ofpropagation andtoeach other.] dV d2V (12)Find asolution of-^-=K^-j- such that dt ox* y=^=ooift=*-foo; V if=0orTT,forallvalues oft; V^TTX-X* if=0,forvalues ofxbetween and TT. [2V.B.Before attemptingthisquestion readagainArts. 46and49. Visthetemperatureofanon-radiatingrodoflengthTTwhose ends are keptat0,thetemperatureoftherodbeing initially (TTX-x2 )ata distance xfromanend.] (13)What does thesolution ofthe lastquestion become ifthe lengthoftherod isIinstead ofTT? [N.B.Proceed asinArt. 50.] (14)Solvequestion (12)ifthecondition F=0 forx=orTTis dV replaced by^--=forx=>Q orTT. [Instead oftheendsbeingataconstant temperature, theyarehere treated sothatnoheatcanpassthrough them.] (15)Solvequestion (12)iftheexpressionTTX-x2isreplaced by100. 60 DIFFERENTIAL EQUATIONS 97 (16)Find asolution of-^-^K^-^such that F=/=ooift=*+00; F100 ifz=orTTforallvalues of*;F0 if forallvalues ofxbetween and TT. [Here theinitially ice-cold rodhas itsends inboiling water.] (17)Solvequestion (15)ifthelengthisIinstead ofTT. IfIincreases indefinitely, show that theinfinite series becomes theintegral 200riK, t. , - j-e~Katsmaxda. TTJo [N.B.This iscalled aFourier'sIntegral. Toobtain this result put (2r+l)7r/J=aand27r//=da. Kelvin usedanintegralinhiscelebrated estimate oftheageofthe earth from theobserved rate ofincrease oftemperature underground. (Seeexample (107)ofthemiscellaneous setattheend ofthebook.) Strutt's recent discovery thatheat iscontinually generatedwithin the earth byradio-activeprocesses shows that Kelvin's estimate wastoo small.] dV 92F (18)Find asolution of-=-=#5-=- such that at ox* Visfinitewhen t=+oo; 9F ^--=0 when x=0,.... _ ex j-forallvalues oft; F=whenx^lj F=*Fwhen t=0,forallvalues ofxbetween and I. [Ifasmall test-tubecontainingasolution ofsalt iscompletely submergedinavery largevessel fullofwater, thesaltdiffuses upout ofthetest-tube intothewater ofthelargevessel. IfFisthe initial concentration ofthesaltand Ithelengthoftest-tube itfills,Vgives theconcentration atanytime ataheightxabove thebottom ofthe 97 test-tube. Thecondition ^=when z=means thatnodiffusion9# takesplaceattheclosed end.F=when x=lmeans that atthetop ofthetest-tube wehavenearly pure water.] 92V 92v (19)Find asolution of^-~=fl2=r^suchthatxdt2ox2 yinvolves xfcrigonometrically; ^=when x=*Q orTT,forallvalues oft; ~=0 when J=0, forallvalues ofx; ot y**mx between z=and-, I \when t=*0. ym(TT-x)between x=-^and TT,I MISCELLANEOUS EXAMPLES 61 [N.B. Seethesecond worked exampleofArt. 48. yisthetransverse displacementofastring stretched between two pointsadistance TTapart. Thestringispluckedaside adistance m7r/2atitsmiddle point andthen released.] *(20)Writing thesolution ofj-|=*Z)2 y,whereDisaconstant, in theform y-^A+e-'iB, deduce thesolution of=-?=-vintheform bysubstituting^-forJD,f(t)and F(t)forAand5respectively, and using Taylor's theorem initssymbolical form [The results obtained bythese symbolical methods should be regarded merelyasprobablycorrect. Unless theycanbeverified by other means, averycareful examination oftheargumentisnecessary tosee ifitcanbetaken backwards from theresult tothedifferential equation. Heaviside hasused symbolical methods tosolve some otherwise insoluble problems. SeehisElectromagnetic Theory.] *(21 )From thesolution of-~=Z)2 y,whereDisaconstant, deduce o ~\9 dx that of3=5-^intheformdx dt224 [Thisisnotasolution unless theseries isconvergent.] General solution of|^=i |^f. Asatrial solution putyf(x+mt),wheremisconstant. m2 Thisgivesf(x+mt)=^.f(x+mt), which issatisfied ifm= 0. Thusyf(x-at) andy=F(x+at)aretwo solutions, andasthe differential equationislinear, athird solution is y=/($_at)+F(x+a/),. containinganumber ofarbitrary functions equal totheorder (two) ofthedifferential equation,sonomore generalsolution canbeexpected. (Of.pp.218and256.) [Arts. 178-181 form asupplement tothischapter. They dealchiefly with theequation ofvibrating strings andwith thethree-dimensional wave equation. Attheend ofArt.181 isalistofsome important works onthedifferential equationsofMathematical Physics.] *Tobeomitted onafirstreading. P.D.B. V CHAPTER V EQUATIONS OFTHEFIRST ORDER BUTNOTOFTHE FIRST DEGREE 51.Inthischapter weshall dealwithsomespecial typesof equationsofthe firstorder and ofdegree higherthan the first for which thesolution cansometimes beobtained without theuseof infinite series. Forbrevity dy/dxwillbedenoted byp. Thesespecial typesare : (a)Those solvable forp. (b)Those solvable fory. (c)Those solvable forx. 52.Equations solvable forp.Ifwecansolve forp,theequation ofthenth degreeisreduced tonequationsofthe firstdegree,to which weapplythemethods ofChap.II. Ex.(i).Theequation p2+px+py +xy=0 gives p=*-x orp=-y ; fromwhich 2y=-x2+cxorx=-logy+ca; or,expressedasoneequation, (2y+z2-c1)(z+log</-c 2)=0 (1) Atthispointwemeet withadifficulty ;thecomplete primitive apparentlycontains twoarbitrary constants, whereas weexpect only one,astheequationisofthe firstorder. Butconsider thesolution (2t/+z2-c)(z+logy-c)=0 (2) Ifweareconsidering onlyonevalue ofeach oftheconstantsc,clf and c2,theseequationseach representapairofcurves, andofcourse notthesamepair (unlessc=c1=c2).But ifweconsider theinfinite setofpairsofcurves obtained bygiving theconstants allpossible values from-ooto+oo,weshallgetthesame infinite setwhen taken altogether, though possiblyinadifferent order. Thus(2)canbetaken asthecomplete primitive. EQUATIONS OFTHEFIRSTORDER 63 Ex.(ii). ;?2+7>-2=0. Here p=>1orp--2, giving f/^z+Cjort/=>-2x-fca. Asbefore, wetake thecomplete primitive as not (y-as- Each oftheseequations representsalllinesparalleleither to y=xortoy-2x. Examples forsolution, (1) />2+p-6=0.(2)p*+2xp~3x*. (3) jo2=x5 . (4)x-f2/pa=p(l+#?/). (5)^3-p(^2+^2/+2/2)+jr2/(a;+2/)==0. (6)p2-2pcosh a?-f1=0. 53.Equations solvable fory.Iftheequationissolvable fory, wedifferentiate thesolved form withrespecttox. Ex.(i). p2-py-fa?=0. Solvingfory, y=p+-* vv/v ,- dp Ixdp Differentiating, p=-+---o~r & rdxpp2dx .6. 1V I7\n p/dpp2 This isalinearequationofthe first order, considering pasthe independentvariable. ProceedingasinArt. 19,thestudent willobtain \~4 x i Hence, asy**p+- 9y=*p+(c+cosir1??)^2--!). These twoequationsforxandyinterms ofpgivetheparametric equationsofthesolution ofthedifferential equation.Foranygiven value ofc,toeach value ofpcorrespondonedefinite value ofxand oneofy,definingapoint. Aspvaries, thepoint moves, tracing out acurve. Inthisexample wecaneliminate pandgettheequationcon- necting xandytbutfortracingthecurve theparametricforms areas good,ifnotbetter. Ex.(ii). ap5-;py+l=0. Solvingfory, y=3^4-fp~l . Differentiating, p-12p3~-p~*~,~-~, i.e.dx=(l2p2-p~z)dp. Integrating,x=4^34-\p~2+c, andfrom above, y=3^4+^r1 . Thestudent should trace thegraphofthisforsome particularvalue ofc,sayc=0. 64 DIFFERENTIAL EQUATIONS 54.Equations solvable for x.Iftheequationissolvable forx, y wedifferentiate thesolved form withrespecttoy,andrewrite --,- ^dy intheform-. P Ex.p2-py+%=0.Thiswassolved inthelast article bysolving fory. Solvingforx, x=py- p1 , Differentiating with respecttoy, I dp_dp~~>p+y^--2p^~,p*ydy^dy p/dp^y"** which isalinear equationofthe first order, considering pastheinde- pendent andyasthedependent variable. Thismaybesolved asin Art. 19.Thestudent willobtain theresult found inthelast article. Examples forsolution. (1)a;=4^-f4^8 .(2)p*-2o^+l=0. (3)y~p*x+p. (4) (5)p*+p=ey.(6) (7)p*-p (y-}-3)4-a;=0.(8)y=psinp+cosp (12)Prove that allcurves ofthefamily given bythesolution of Ex. 1cuttheaxis ofyatright angles. Find thevalue ofcforthat curve ofthefamilythatgoesthrough thepoint (0,1). Trace thiscurve onsquared paper. (13)Trace thecurve given bythesolution ofEx.9with c=0. Draw thetangentsatthepoints given byp=Q,p=*'I, p='2andp=-3, andverify, bymeasurement, that thegradientsofthesetangentsart* respectively 0,*1,*2and *3. CHAPTER VI SINGULAR SOLUTIONS* 55.Weknow from coordinate geometrythat thestraightline y=*mx +touches theparabola y2=4ax,whatever thevalue ofm. Consider thepointofcontact Pofanyparticular tangent. At Pthetangent andparabolahave thesame direction, sotheyhave acommon value of^-,aswellasofxandy. FIQ. 7. But forthetangent m=>~=>p say,sothetangentsatisfies the ctx differentialequation y-px+-. Hence theequationholds also fortheparabolaatP,where x, y,andparethesame asforthetangent. AsPmaybeanypoint ontheparabola,theequationoftheparabola j/2=4a#must bea solution ofthedifferentialequation,asthestudent willeasily verify. *Thearguments ofthischapterwillbebased upon geometricalintuition. The results therefore cannot beconsidered tobeproved, butmerely suggestedas probablytrue incertain cases. Theanalytical theory presents gravedifficulties (seeM.J.M.Hill, Proc. Lond. Math. Soc.t1918). 65 66 DIFFERENTIAL EQUATIONS Ingeneral,ifwehaveanysinglyinfinitesystemofcurves which alltouch afixed curve, whichwewillcalltheirenvelope* and ifthis family representsthecomplete primitiveofacertain differential equationofthe first order, then theenvelope representsasolution ofthedifferentialequation.Foratevery pointoftheenvelope x,y,andphave thesame value fortheenvelope andthecurve of thefamily thattouches itthere. Such asolution iscalled aSingularSolution. Itdoes not contain any arbitrary constant, and isnotdeducible from the Complete Primitive bygivingaparticularvalue tothearbitrary constant init,save inexceptionalcases(Art. 160). Example forsolution. Prove that thestraightlineyxistheenvelopeofthefamilyof parabolas t/=x+J(x-c)2 .Prove that thepointofcontact is(c,c), andthatp=*lfortheparabola andenvelopeatthispoint. Obtain the differential equationofthefamilyofparabolasintheform y=x+(p-1)2 ,andverify thattheequationoftheenvelopesatisfies this. Trace theenvelope andafewparabolasofthefamily, takingcas 0,1,2,etc. 56.Weshallnowconsider how toobtainsingularsolutions. It hasbeenshown thattheenvelopeofthecurvesrepresented bythe complete primitive givesasingular solution, soweshallcommence byexaminingthemethod offinding envelopes. Thegeneralmethod tistoeliminate theparametercbetween f(x, y,c)=0,theequationofthefamilyofcurves, and E.g.if/(#, y,c)= isy-cx-~=0,.....................(1)c |-0is-X+J.-0,.....................(2) givingc= *InLamb's Infinitesimal Calculus, 2nd ed., Art. 155,theenvelopeofa familyisdefined asthelocus ofultimate intersection ofconsecutive curves of thefamily. Asthusdefined itmayinclude node- orcusplociinaddition toor instead ofwhatwehave called envelopes. (We shallgiveageometrical reason for thisinArt.66 ;seeLamb forananalytical proof.) Lamb'sInfinitesimal Calculus, 2nd ed., Art. 156. Iff(x, y,c,)isof theform Lc* -fMc+N,theresult comes toM*=4LN. Thus, for theresult is y*=4a;. [Arts. 155-156, 2nd ed.,become Arts. 138-139 inthe3rded.] SINGULAR SOLUTIONS 67 Substitutingin(1), y-2V#. or y2=4x. Thismethod isequivalenttofindingthelocus ofintersection of f(x, y,c)=0, andf(x, y,c+A)=0, twocurves ofthefamilywithparametersthat differ byasmall quantity h,andproceedingtothelimitwhen happroacheszero. Theresult iscalled thec-discriminant off(x, y,c)0. 57.Now consider thediagrams 8,9,10,11. Fig.8shows thecasewhere thecurves ofthefamily have DOspecial singularity.The locus oftheultimate intersections m a, PQRSTUVisacurve which hastwopointsincommon witheach ofthecurves ofthefamily (e.g.QandRlieonthelocusandalso onthecurve marked2).Inthelimit thelocusPQRSTUV there- foretouches eachcurve ofthefamily, and iswhatwehave defined astheenvelope. InFig.9each curve ofthefamilyhasanode. Two con- secutive curves intersect inthreepoints (e.g.curves 2and3inthe points P,Q,andR). Thelocus ofsuchpointsconsists ofthree distinctpartsEE'9 AA',andBB'. When weproceedtothe limit, takingtheconsecutive curves ever closer and closer, AA!andBB' willmove uptocoincidence withthenode-locus NN', while EE' willbecome anenvelope.So 68 DIFFERENTIAL EQUATIONS inthiscaseweexpecttheodiscriminant tocontain thesquareof theequationofthenode-locus, aswellastheequationoftheenvelope. B1 AsFig.10shows, thedirection ofthenode-locus NN' atany pointPonitisingeneralnotthesame asthat ofeither branch of thecurve withthenode atP.Thenode-locus hasxandyincommon with thecurve atP,butnotp,sothenode-locus isnotasolutionof thedifferential equation ofthecurvesofthefamily. FIG. 10. Ifthenode shrinks intoacusp,thelociEE'andNN' ofFig.10 move uptocoincidence, formingthecusp-locus CC' ofFig.11. NowNN'wasshown tobethecoincidence ofthetwo lociAA'and BB' ofFig. 9,soCC' isreally thecoincidence ofthree loci,and itsequation must beexpectedtooccur cubed inthec-discriminant. Fig.11shows that thecusp-locus,likethenode-locus,isnot (ingeneral)asolution ofthedifferentialequation. C-T . o' Tosumup,wemay expectthec-discriminant tocontain ; (i)theenvelope, (ii)thenode-locussquared, (iii)thecusp-locuscubed. SINGULAR SOLUTIONS 69 Theenvelopeisasingular solution, butthenode- andcusp- lociarenot(ingeneral *)solutions atall. 58.Thefollowing exampleswillillustrate theprecedingresults : Ex.(i). y=p2 . Thecomplete primitiveiseasily found tobe4z/=(3-c)2 , i.e. c2-2cx+x2-4:yQ. Asthis isaquadraticinc,wecanwritedown thediscriminant at once as(2z)2=4(x24w), i.e.t/=0,representingtheenvelopeofthefamilyofequal parabolas givenbythecomplete primitive,andoccurringtothe firstdegree only, asanenvelopeshould. FIG. 31. Ex.(ii). Proceedingasinthelastchapter, weget i.e. .e.^dp 3orP-2&-J-.rdx dxOdp.(A) *Wesayingeneral, because itisconceivable that insomespecial example a node- oroutp-loous maycoincide withanenvelopeorwithacurve ofthefamily. 70 DIFFERENTIAL EQUATIONS logx=2logp-log c, whence3y=>2c*cc*-2c, i.e.(3y-l-2c)*=4ca^, afamily ofsemi-cubicalparabolas with theircusps ontheaxis ofy. Thec-discriminant ia (3y-x3 )2=9i/a , Thecusp-locus appears cubed, andtheother factorrepresents the envelope. Itiseasily verified thatGy^z3isasolution ofthe differential equation, whilez=0(giving p=oo )isnot. Ifwetake the first alternative oftheequations (A), i.e.x2-2p=0, wegetbysubstitution forpinthedifferentialequation i.e.theenvelope. This illustrates another method offinding singular solutions FIG. 13. Examples forsolution. Find thecomplete primitives andsingularsolutions(ifany) ofthe followingdifferentialequations. Trace thegraphsforExamples1-4: (1)4pa-9a;=0.(2)4j92(-2)=l. (3)xp*-2yp+4x*=0.(4)p2+t/2-l-=0. (5)p*+2xp-y**0. (6) (7) SINGULAR SOLUTIONS 71 59.The p-discriminant. We shallnow consider how toobtain thesingularsolutions ofadifferentialequation directly from the equation itself, without havingtofindthecomplete primitive. Consider theequation x2p2-yp-f1=0. Ifwegivexandyanydefinite numerical values, wegetaquad- ratic forp.Forexample,if 3-y% y^y 2p2-3p+l=0, p=|or 1. Thus there aretwocurves ofthefamily satisfyingthisequation through every point.These twocurves willhave thesametangent atallpoints where theequationhasequalroots inp,i.e.where thediscriminant y2-4o;2=0. Similar conclusions hold forthequadratic Lp*+Mp+N=*Q y where L,M,Nareanyfunctions ofxandy.There aretwocurves through every pointintheplane,butthese curves have thesame direction atallpointsonthelocusM2-kLN=0. Moregenerally,thedifferentialequation f(x,yyp)sLQpn+LlPn-~l+L2p-2+...+Ln=0, where theL'aarefunctions ofxandy,givesnvalues ofpfora given pairofvalues ofxandy,correspondingtoncurvesthrough any point. Two ofthese ncurves have thesametangentatall pointsonthelocusgiven byeliminating pfrom forthis isthecondition giveninbooks ontheoryofequationsfor theexistence ofarepeatedroot. Wearethus ledtothep-discriminant, andwemustnow in. vestigatethepropertiesofthelocirepresented byit. 60.TheEnvelope. Thep-discriminantoftheequation or is y=z. Wehave already found thatthecomplete primitive consists of thetangentstotheparabola,which isthesingularsolution. Two ofthese tangents passthrough every pointPintheplane,and thesetangentscoincide forpointsontheenvelope. 72 DIFFERENTIAL EQUATIONS This isanexampleofthe^-discriminant representinganenvelope, Fig.15shows amoregeneralcase ofthis. Fio. 14. Consider thecurveSQP asmoving uptocoincidence with the curvePRT, always remainingincontact with theenvelope QRU. ThepointPwillmove uptowards R,andthetangentstothetwo curvesthrough Pwillfinallycoincide witheach other andwiththe tangent totheenvelopeatR.ThusRisapointforwhich thep's ofthetwocurves ofthesystem throughthepoint coincide, and consequentlythe^-discriminantvanishes. U PIG. 15. Thus thep-discriminant maybeanenvelopeofthecurves of thesystem, and ifso,asshown inArt. 55,isasingularsolution. 61.The tac-locus. Theenvelopeisthus thelocus ofpoints where two consecutive curves ofthefamily have thesame value ofp.But itisquite possiblefortwonon-consecutive curves to touch. Consider afamilyofcircles, allofequal radius, whose centres lieonastraightline. SINGULAR SOLUTIONS 73 Fig.16shows thatthelineofcentres isthelocus ofthepoint ofcontact ofpairsofcircles. This iscalled atac-locus. Fig.17 E E' FIG. 16. shows circles which donotquite touch, butcutinpairsof bouring points, lyingontwoneighbouringlociAA'yBB'. When' weproceedtothelimitingcase ofcontact thesetwo locicoincide inthetac-locns TT '.Thus the^-discriminant maybeexpectedto contain theequationofthetac-locussquared. FIG. 17. Itisobvious that atthepointPinFig.16thedirection of thetac-locus isnotthedirection ofthetwo circles. Thus the relation between x,y,andpsatisfied bythe circles willnotbe satisfied bythetac-locus, which hasthesame xandybutadifferent patP.Ingeneral,thetac-locus does notfurnishasolutionofthe differential equation. 62.The circles ofthelast article arerepresented by (x+c)2-fy2=ra , ifthelineofcentres isOx. Thisgivesx+c=>Vr2- t/2 , or I--yplVr*-y\ i.e.*/2y2+y2-r2=0. Thejo-discriminantofthis isy2 (y2-r2 )=0. The liney=0 (occurring squared,asweexpected)isthetac- locus, y=dtraretheenvelopesEE'andFF' ofFig.16; ?/=r, giving p=0,aresingularsolutions ofthedifferentialequation,but y=doesnotsatisfyit. 63.The cusp-locus. Thecontact thatgivesrisetotheequal roots inpmaybebetween twobranches ofthesame curve instead 74 DIFFERENTIAL EQUATIONS ofbetween twodifferent curves,i.e.thep-discriminantvanishes at acusp. Asshown inFig. 18,thedirection ofthecusp-locusatany pointPonitisingeneralnotthesame asthat ofthetangentto thecusp,sothecusp-locusisnotasolutionofthedifferential equation. C' Pro. 18. Itisnatural toenquireiftheequationofthecusp-locuswill appearcubed inthep-discriminant,asinthec-discriminant. To decide this, consider thelocus ofpointsforwhich thetwop'sare nearlybutnotquite equal, when thecurves haveveryflatnodes. This willbethelocusNN' ofFig.19. Inthelimit,when thenodes FlO. 19. contract intocusps, wegetthecusp-locus, andasinthiscasethere isnoquestionoftwo ormore locicoinciding, weexpectthep- discriminant tocontain theequationofthecusp-locustothe first power only. 64.Summary ofresults. Thep-discriminanttherefore maybe expectedtocontain (i)theenvelope, (ii)thetac-locussquared, (iii)thecusp-locus, andthec-discriminant tocontain (i)theenvelope, (ii)thenode-locussquared, (iii)thecusp-locus cubed, SINGULAR SOLUTIONS 75 Oftheseonlytheenvelopeisasolution ofthe differential equation. 65.Examples. Ex.(i). j>*(2-3*,)^4(l-y). Writingthisintheform dx 2-3 weeasilyfindthecomplete primitiveintheform Thec-discriminant and^-discriminantarerespectively !/2(l-y)=0 and (2-3</)2(l-t/HO. 1_yr=Q,which occurs inboth tothefirstdegree, givesanenvelope; /=0,which occurs squaredinthec-discriminant andnotatallin the^-discriminant, givesanode-locus;2-3?/=0,which occurs squared inthep-discriminant andnotatallinthec-discriminant, givesa tac-locus. Ifciseasilyverified that ofthese three locionlytheequationofthe envelopesatisfies thedifferentialequation. Tac-locusFhvetope Node-Jocus FIG. 20. Ex.(ii).Consider thefamilyofcircles Byeliminatingc(bythemethods ofChap. L),weobtain thediffer- ential equation 2y2 2>2+2xyp+za+y*-1~< 76 DIFFERENTIAL EQUATIONS The c-and^-discriminantsarerespectively z2-2(z2+t/2-l)=0 and2%2-2?/2 (:r2+ %v2-l)==0, i.e.x2-f2t/2-2=and y2 (x2+2y2-2)=0. 2+2?/2-2=0gives anenvelopeasitoccurs tothe firstdegreein bothdiscriminants, whilef/=givesatac-locus, asitoccurs squared inthe^-discriminant andnotatallinthec-discriminant. The circle given bytheoriginal equationtouches theenvelopeatthepoints {-2 C)s/(l-2c2 )}, which areimaginary when cisnumerically greaterthan|\/2. FIQ. 21. Examples forsolution. Inthefollowing examplesfindthecomplete primitiveifthediffer- entialequationisgivenorthedifferentialequationifthecomplete primitiveisgiven. Find thesingularsolutions(ifany). Trace the graphs. (1)4x(x-~l)(x-2)p*-(3x*-6x +2)*~0. (2) (3)yp*-2xp +y~Q. (4) (5)p*+2px*-4x*y=*Q. (6) (7)z2+y2-2c3+c2cos2a==0.(8) (9)c*+(x+y)c+l-xy=Q. (10)x2+2/2+2cxy+c*-l=0. 66.Clairaut's Form.* sideringtheequationWecommenced thischapter bycon- a ~P% *Alexis Claude Olairaut, ofParis (1713-1765), althoughbestknown incon- nection with differential equations, wrote chiefly onastronomy. SINGULAR SOLUTIONS This isaparticularcase ofClairaut's Form y~px+f(p) Tosolve, differentiate withrespecttox.77 (1) therefore /=0, p=c, (2)dxr or O-s+Hp) (3) Using (1)and(2)wegetthecomplete primitive,thefamilyof straight lines,y=cx+f(c) (4) Ifweeliminate pfrom(1)and(3)weshallsimply getthe^-dis- criminant. Tofindthec-discriminant weeliminate cfrom(4)andtheresult ofdifferentiating (4)partiallywithrespecttoc,i.e. 0-*+/'(c) (5) Equations (4)and(5)differ from(1)and(3)onlyinhavingc instead ofp.Theeliminants aretherefore thesame. Thus both discriminants mustrepresenttheenvelope.* Ofcourse itisobvious that afamilyofstraightlinescannot have node-, cusp-,ortac-loci. Equation (4)givestheimportantresult that thecomplete primi- tiveofadifferential equation ofClairaut's Formmaybewritten down immediately bysimply writingcinplace ofp. 67.Example. Find thecurve such thatOTvaries astan^,whereTisthepoint inwhich thetangentatanypointcutstheaxisofx,^isitsinclination tothisaxis,and istheorigin. y oiN * FIO. 22. *Butinsome cases thediscriminantsrepresent notonlytheenvelope, butalso Itsinflexional tangents (Art. 161). P.D.S. o 78 DIFFERENTIAL EQUATIONS From thefigure, OT~ON-TN -*-*. snce therefore 05-~=Jti)t P* i.e.y=*px-kp*. This isofClairaut's Form, sothecomplete primitiveis y=*cx-kc2 , andthesingularsolution isthediscriminant ofthis, i.e. x*=4%. Thecurverequiredistheparabola represented bythissingular solution. Thecomplete primitive representsthefamily ofstraight linestangenttothisparabola Examples forsolution. Find thecomplete primitive andsingularsolutions ofthefollowing differentialequations.Trace thegraphsforExamples (1), (2), (4), (7), (8)and(9). (1)y=px+p*. (2)y=*px+p*. (3)y=*px+cosp. (4)y=px+\/(a2p*+b*). (5)p=log(px-y). (6)sinpxcosy=cospxsin y+p. (7)Find thedifferential equationofthecurve such thatthetangent makes with theco-ordinate axes atriangleofconstant areaia ,and hence findtheequationofthecurVe inintegralform. (8)Find thecurve such that thetangent cuts offintercepts from theaxeswhosesum isconstant. (9)Find thecurve such that thepartofthetangent intercepted between theaxes isofconstantlength. MISCELLANEOUS EXAMPLES ONCHAPTER VI. Illustrate thesolutions byagraph wheneverpossible. (1)Examine forsingularsolutions (2)Eeduce xyp*-~(x* +y*- toClairaut's formbythesubstitution X=*x*;Fy*. Hence show thattheequation representsafamilyofconiestouching thefour sides ofasquare. MISCELLANEOUS EXAMPLES 79 (3)Show that xyp*+(x*-y*-h*)p-xy*=*() representsafamilyofconfocal conies, withthefociat(A, 0),touching thefourimaginarylinesjoiningthefocitothecircularpointsatinfinity. (4)Show bygeometrical reasoningorotherwise that thesub- stitution x~aX+bY 9y=a'X+VY, converts anydifferential equationofClairaut's form toanotherequation ofClairaut's form. (5)Show that thecomplete primitiveof8p*x=*y(l2p2-$)is (x+c)3=3y2 c,thep-discriminant y2(x*~4y*)=*b, and the c-dis- criminant/4 (9a?a~4y2 )=0.Interpretthese discriminants. (6)Reduce thedifferential equation x2p2+yp(2x+y)+y*=*0> wherep^-jdx toClairaut's formbythesubstitutiony, rjxy. Hence, orotherwise, solve theequation. Prove that y+4#= isasingularsolution;andthatyisboth partoftheenvelope andpartofanordinarysolution. [London. ] (7)Solvey2(y-^Y}^\ijwm^hcanketransformed to Clairaut's formbysuitable substitutions. [London.] (8)Integratethedifferentialequations: (i) (ii) In(ii)findthesingularsolution andexplain thesignificanceofany factors that occur.[London.] (9)Show thatthecurves ofthefamily allhave acuspattheorigin, touching theaxis ofx. Byeliminatingcobtain thedifferential equationofthefamilyin theform 4p2x2(x-1)-4:pxy (4x-3)+(16*-9)?/2-0. Show thatboth discriminants take theformx3y2= Jbutthatz= isnotasolution, while y=isaparticular integral. [Thisexampleshows thatourtheorydoesnotapplywithout modi- fication tofamilies ofcurves with acuspatafixedpoint.] (10)Show thatthecomplete primitiveof representsthefamilyofequallemniscates ofBernoulli ra=a2cos2(0~a), Inscribed inthe circle ra,which isthesingular solution, with the pointr=0asanode-locus. 80 DIFFERENTIAL EQUATIONS (11)Obtain andinterpretthecomplete primitive andsingular solution of/dr\* (m)+r*-2=- (12)Show that r=c0-c2isthecomplete primitive and4r=2the singular solution of ^ /^.\ Verify thatthesingular solution touches thecomplete primitive at thepoint (c2 ,2c),thecommontangentthere making anangle tan""^ withtheradius vector. [For asupplementary discussion ofsingular solutions, including difficulties concerningtheir definition andthedefinition ofanenvelope, theoccurrence ofparticularsolutions inthediscriminants, theidea of boundaries, andthemethods ofcalculating discriminants, seeArts. 160-161. These willthrow additionallightonExs. 7and9above.J CHAPTER VII MISCELLANEOUS METHODS FOREQUATIONS OFTHE SECOND ANDHIGHER ORDERS 68.Inthischapter weshall beconcernedchiefly with the reduction ofequationsofthesecond order tothose ofthe first order.Weshallshow thattheorder canalways besoreduced if theequation (i)doesnotcontain yexplicitly; or(ii)doesnotcontain xexplicitly; or(iii)ishomogeneous. Aspecialform ofequation,ofsomeimportanceinDynamics, maybereduced byusing anintegratingfactor. Theremainder ofthechapterwillbedevoted tothelinear equation, excludingthesimple case, already fullydiscussed in Chapter III.,where thecoefficients aremerelyconstants. Itwill befound thatthelinearequationofthesecond order canbereduced tooneofthe firstorder if (i)theoperatorcanbefactorised, or(ii)anyoneintegral belongingtothecomplementaryfunction isknown. Ifthecomplete complementaryfunction isknown, theequation maybesolved bythemethod ofVariation ofParameters. This elegant method (duetoLagrange)isapplicabletolinearequations ofanyorder. Further information onlinearequations,such asthecondition forexactequations,thenormal form, theinvariantive condition of equivalence, andtheSchwarzian derivative, willbefound inthe form ofproblems amongthemiscellaneous examplesattheend ofthechapter,with hints sufficient toenable thestudent towork them outforhimself. 81 82 DIFFERENTIAL EQUATIONS Weshall usesuffixes todenote differentiations withrespectto dhix*t-9*y%fr jJ 2,butwhen theindependentvariable isanyother thanxthedifferential coefficients willbewritten infull. 69.yabsent. Ifydoesnotoccurexplicitlyinanequationof thesecond order, writepforyland-j-fory2. Weobtain anequation containing only ,^,p,and x,andsoof thefirst order. Consider, forexample, xy2+yl4x. Thistransforms into x~+p=4#,dxr which canbeintegratedatonce xp^2x*+ a, .e.p*x Byintegrating, y=x*+alogx+6, where aand6arearbitraryconstants. Thismethod maybeused toreduce anequationofthentkorder notcontaining yexplicitlytooneofthe(n-l)th . 70.xabsent. Ifxistheabsent letter, wemaystillwritepfor y>,butfory,wenowwrite j>|,sincep&-g|-g- j,,.The procedurereduces anequationofthesecond order without xtoone ofthe firstorder inthevariables pandy. Forexample, yy2=y^ transforms intoyp~=pa , fromwhich thestudent willeasilyobtain =*b and y Examples forsolution. (1)y,cos1s-1.(2)yfc+y!1-^. (3) (4)Reduce totheprevious example,andhence solve (5)a?y8+y,12a?. (6)yn-2yn (7)Integrate andinterpret geometrically EQUATIONS OFSECOND ANDHIGHER ORDERS 83 (8)Theradius ofcurvature ofacertain curve isequaltothelength ofthenormal between thecurve andtheaxis ofx.Prove that the curve isacatenaryoracircle, accordingasitisconvex orconcave to theaxis ofx. (9)Findandsolve thedifferential equationofthecurve thelength ofwhose arc,measured from afixed pointAtoavariablepointPyis proportionaltothetangentoftheangle between thetangentatPand theaxis ofx. *71.Homogeneous equations. Ifxandyareregardedasof dimension 1, yvisofdimension 0, y2isofdimension -1, j/3isofdimension -2, andsoon. Wedefine ahomogeneous equationasoneinwhich alltheterms areofthesame dimensions. WehavealreadyinChap.II.dealt withhomogeneous equationsofthe firstorder anddegree, and in Chap.III.with thehomogeneouslinearequation xnyn+Axn-lyn_i+Bxn-*yn^-f...+IIxy l+Ky=0 (where A,B,...Z7,Karemerely constant?),forwhich weused the substitution xe(ort=logx. Letusmake thesame substitution inthehomogeneous equation VVV2 +22/i2=%2/i............................(1) dtdy 1dy ^TNow dyl l^dy1ddy 7fo" x*dt xdx dt y_I y x2dtxdxdt* _" x*dt x*dt*' Substitutingin(1)andmultiplying byx,weget This isanequation, with tabsent, similar tothose inthelast article withxabsent. *Arts. 71-73maybeomitted onafirstreading. 84 DIFFERENTIAL EQUATIONS n1J Byputting-~=y,thestudent willeasilyobtain gvng Hence y2+6=e4(t+c) =ax4 ,replacinge*cbyanotherarbitraryconstant a. 72.TheexampleofArt. 71came outeasily because ithadno superfluousx'sleftafterassociating x2withy%andxwithyx.In fact, itcould havebeen written But(*2+y2)(y-^i)+zyy 2-0 .....................(2) cannot besowritten. Toreduce thistoaform similar tothat of thelastexample, puty=vx,asubstitution used forhomogeneous equationsinChap.II. (2)becomes (x2+x2v2 )(vx-i\xz-vx)+x*v2(xv2+2^)=0, i.e.-(1+V2 )vl+v2(xv2+2vl)-0, whichmaybewritten v*x2v2=(I-v2}xvl............................(3) Wenowproceedasbefore andputx=ef ,giving dvxv^df .,d2vdvand **>*--' ,n.i 9fd2vdv (3)bccome3 v ~ dv anequationwith tabsent. . ,, , dv d2vdaAsbefore, put Tr^tf* jt9~Q -j*tit dt (it) (4)becomes v2qy*=j, i.e. =(unless gr=0,giving y-cx) t dv11 _avdv/a2\-, fa a ffl-f)rfV,v-a \v-a/ andfinally logxay/x+a2log(y-ax)-a2loga?+6. EQUATIONS OFSECOND ANDHIGHER ORDERS 85 73.Byproceedingasinthe last article, wecanreduce any homogeneous equationofthesecond order. Anysuchequation canbebroughttotheform Forexample,theequationofArt.71when divided byxbecomes while that ofArt.72divided byx3becomes HDCS)*- Thesubstitutions y=>vxandx^e1transform y*><%)- to/(vyxvl+v,x2i\2+2xi\)-0, 1,1 i f(dv d2vdvandthen tof(v>dt+v>d^+ anequationwith tabsent, andtherefore reducible tothe firstorder Examples forsolution. (I)x*yt-xy 1+y~Q. (2)X2y2-xy 1+6y-0. (3)2x2 //,v2+*/=z5V. (4)Make homogeneous bythesubstitution y~z*, andhence solve 74.Anequation occurring inDynamics. Theformy^ occursfrequentlyinDynamics, especiallyinproblemsonmotion under aforce directed toafixedpointand ofmagnitude depending solely onthedistance from that fixedpoint. Multiplyeach side oftheequation by2yL.Weget f dii t Integrating, yf=2/(y)fdx-2/(y)dy. J HiJu J This isreallytheequationofenergy. Applyingthemethod to ,a-p*x, (theequationofsimple harmonic motion), weget odxdzxo2dx Integratingwithrespectto, (dx2 J=~p2x2+const.= j>2(a2-cr2 86 DIFFERENTIAL EQUATIONS TT"==3 " da?p ,1i =-sin"1-+const.. > a xasn Examples forsolution. (!)y*=*f-y> giventhat^=when y-1. (2) t/2=e2y ,giventhat?/=andy^l when x=0. (3)y2=>sec2 2/tany,giventhaty^Oandt/i^l when z=0. WJT888~" agiventhatx=handy-=0when t=Q. (tt x dt [h-xisthedistance fallen from restundergravity varying inversely asthesquareofthedistance xfrom thecentre oftheearth, neglecting airresistance, etc.] (5)i2a+M=aL2~~ainthetwo cases giventhat==0when =-,wherejm,h,and careconstants. ct(/ c [These givethepath described byaparticleattracted toafixed pointwithaforcevarying inverselyasthesquare andcuberespectively ofthedistance r.uisthereciprocalofr,6has itsordinary meaning inpolar co-ordinates, JJListheacceleration atunit distance, andhis twice thearealvelocity. ] 75.Factorisation oftheoperator. Thelinearequation (x maybewritten as whereDstands forj-,asinChapterIII. Now theoperatorinthisparticular examplecanbefactorised, giving{( Put Then This isalinearequationofthefirstorder. SolvingasinArt. 20, weget v=c(jc+2) +e*, i.e.(D-2)y=c(x+2)+e* 9 another linearequation, giving finally ya(2x+5)+be2x-ex ,replacing-Jcbya. EQUATIONS OFSECOND ANDHIGHER ORDERS 87 Ofcourse itisonlyinspecialcases that theoperatorcanbe factorised. Itisimportanttonotice that these factors must be written intheright order, astheyarenotcommutative. Thus, on reversingtheorder inthisexample, weget (D-2){(x+2)D-l}y={(x+2)D2-(2x+4)D+2}y. Examples forsolution. (1)(x+l)y1+(a5-l)y 1-2y-0. (2)xyt+(x-l) yi-y-0. (3)xyz^(x-l)y l-y^x\ (4) xt/a+(a52+l)y l+2ajt/==2ir, giventhat t/<=2 and^1=aOwhen (5)(x2-1)7/2-(4z2-3x-5)yl+(4x2-6a?-5)y=e2 *,giventhaty-1 and2/i=2when x=0. 76.One integral belonging tothecomplementary function*known. When oneintegraloftheequation y*+Pyi+Qy~Q..............................(i) isknown, sayy^z, then themoregeneral equationofthesecond orderyi+fyi+Qy-fl,..............................(2) where P,Q,Rarefunctions ofx,canbereduced tooneofthe first orderbythesubstitutiony=*vz. Differentiating, y**vz+vzl9 Hence(2)becomes vf+%(2^+Pz)+v(za+Pzl+Qz)=7?, i.e. z( !j+vl(2z1+Pz)~R,.........................(3) sincebyhypothesisz2+Pzl-fQz=0. (3)isalinearequationofthe firstorder int^. Similarlyalinearequationofthent!lorder canbereduced to oneofthe(n-l)thifoneintegral belongingtothecomplementary function isknown. 77.Example. Consider againtheeqiation (4) *The proofofArt.29that thegeneralsolution ofalineardifferential equation M thesumofaParticular Integral and theComplementary Function holds goodwhen thecoefficients arefunctions ofxaswellasinthecasewhen theyareconstants. 88 DIFFERENTIAL EQUATIONS Ifwenotice thaty=*e2xmakes theleft-hand side oftheequation zero,wecanput yVe2* 9 giving yl= ( and2/2^ ( Substitution in(4)gives (x+2)v2e*x+(4(x+2)-(2x+5)}v^e Solvingthis intheusualway(byfindingtheintegrating factor) weobtain ^e-x+c(x+2)<r2x . Integrating, v=-cr* -Jc(2a;+5)e~2a3-f6, whence y=t'e2*==-ex-\c(2x+5)+6e2a} . Examples forsolution. (1)Show thatyss+P^+Qy-Oissatisfied by t/=e*if1+P+G-O, andby^=x-if (2) (3) (4) (5)x2 z/2+^i~9?/=0, given that2/=o^isasolution. (6)xy2-(xcosx-2 sinx)-f(a?2+2)2/ 1sin-2y(xsin x-fcosx}0, giventhatT/=x2isasolution. 78.Variation ofParameters. Weshallnowexplainanelegant butsomewhat artificial method forfindingthecomplete primitive ofalinearequation whosecomplementaryfunction isknown. Letusillustrate themethod byapplyingittotheexample alreadysolved intwodifferentways, namely, (+2)y 1-(2aj+5)y 1+2y-(a; +l)c,..................(1) ofwhich thecomplementaryfunction isy=a(2x-f-5) +be2x . Assume that y~(2x+5)A +e2xB,...........................(2) whereAandBarefunctions ofx. Thisassumptionissimilar to,butmoresymmetrical than, that ofArt. 77,viz. :y=-vezx . Differentiating (2), yl=(2x+5)A l-}-e2xB1+2A+2eZxB...................(3) Now sofarthetwofunctions(orparameters) AandBareonly connected byasingle equation. Wecanmake themsatisfy theadditional equation ttx+5)^+ e^Bt-0................ ....... (4) EQUATIONS OPSECOND ANDHIGHER ORDERS 89 (3)willthenreduce to ...............................(5) Differentiating (5), t/2=462*B+241+2e2*B1.........................(6) Substitute these values ofy,yvandy2fromequations (2), (5), and(6)respectivelyin(1).The co-factors ofAandBcome to zero, leaving 2(z+2)4 1+2(z+2)62*S1==(o;+l)ea!.................(7) (4)and(7)aretwosimultaneousequationswhich wecansolve forAlandBltgiving 4lgi___ e2*"- (<2x+5) ~ .(s+l)e*e3 '/1Hence^--- -- |2 6* and,byintegration, A=--r- f- o\+a>where aisaconstant. 4:(X+2ij Similarly, E(^+5X^+1)6^^6^1 __ l___1\1==4(x+2)24I a?+2(z+2)2/' and JB- Substitutingin(2), 79.Applyingtheseprocessestothegenerallinearequationof thesecond order,y*+Pyi+Qy-R,...........................(1) ofwhich thecomplementary function au+bv issupposed known, aandbbeing arbitraryconstants anduandvknown functions of#, weassume thaty=uA+vB, ..............................(2) giving ft-w^+t^JB,...........................(3) provided that uA1+vBl=Q............................(4) Differentiating (3), y2^u2A+v2B+u1A1+v1Bl......................(5) Substitute fory2,ytandyin(1). Thetermsinvolving AwillbeAfa+Pu^+Qu),i.e.zero, asby hypothesis, u^+pu+QU~Q. Similarlythetermsinvolving Bvanish, and(1)reduces to R...............................(6) 90 DIFFERENTIAL EQUATIONS Solving (4)and(6),*~A--?_&v' v'V-U VUi-UV^* WethengetAandBbyintegration, say A=f(x)+a, B~F(x)+b, where/($)andF(x)areknown functions ofx,andaand6are arbitraryconstants. Substitutingin(2),wegetfinally y-uf(x)+vF(x)+au+ bv. *80.Thismethod canbeextended tolinear equationsofany order. Forthat ofthethird order, y*+Py*+Qyi+Ry-8,.......................... (i) ofwhich thecomplementaryfunction y=au+bv+cw issupposed known, thestudent willeasilyobtain theequations .............................. (2) ,........................... (3) providedthat 0^uAl-^vBl+wCl;........................... (4) hence y2=u2A+v2B+w2C,........................... (5) providedthat 0=u1A1+v1B1-^-wlC1;........................(6) then* ys=u3A+vJS+wsO +u2Al+v2Bl+w2C1;........................ (7) bysubstitution in(1),8^fa2Al+v2Bl-\-w2Cl......................... (8) A19B19andClarethenfound from thethreeequations (4), (6/ and(8). Examples forsolution. (4)xzy2-f-xy-y=x*e*,giventhecomplementaryfunction ax+bar1* 81.Comparison ofthedifferent methods forsolving linear equations. Ifitisrequiredtosolve alinearequationofthesecond order and nospecial method isindicated, itisgenerallybest totrytoguess aparticular integral belongingtothecomplementaryfunction and proceedasinArt. 76.Thismethod maybeused toreduce alinear equationofthewthorder tooneofthe(n-l)tb . *Tobeomitted onafirstreading. EQUATIONS OFSECOND ANDHIGHER ORDERS 91 Themethod offactorisation oftheoperator givesaneat solution inafewcases, butthese areusually examples speciallyconstructed forthispurpose. Ingeneraltheoperator cannot befactorised. Themethod ofvariation ofparametersisinferior inpractical value tothat ofArt. 76,asitrequiresacomplete knowledgeofthe complementaryfunction instead ofonlyonepartofit.Moreover, ifappliedtoequationsofthethird orhigher order, itrequirestoo much labour tosolve thesimultaneousequationsforA19J3,,Cltetc., andtoperformtheintegrations. MISCELLANEOUS EXAMPLES ONCHAPTER VII. (!) 2/2/2- 2/i2+2/i-0.(2)a^+zt/!2-2/i=0. (3) 2/2-tyn-r (4)yn+*/M_2=8costo. (5) (6) (7)Verify that cosnxand sinnxareintegratingfactors of Hence obtain two firstintegralsof ?/2+M2 2/=secnx, andbyelimination ofyldeduce thecomplete primitive. (8)Show thatthelinearequation where A,B,C,...Tarefunctions ofxyisexact,i.e.derivable imme- diately bydifferentiation fromanequationofthenextlower order,if thesuccessive differential coefficients ofA,B,C,...satisfy therelation ^-J?1+1-...+(-l)S n-0. [N.B. Bysuccessive integration byparts, {Sy.cZ*=%,_!-Siyn_2+Styn_a+...+(- l)-S, l_1y+J(-l)S nyda,.] Verifythat thiscondition issatisfied bythefollowing equation,and hence solve it : (9)Verifythat thefollowingnon-linearequationsareexact, and BO!vethem : (i)^g+y^_,Q. (ii)xyyt+xy^+yy^Q. (10)Show thatthesubstitution y=>veJ transforms yt+Pyi+Qy~R, where P,Q,andRarefunctions ofx,intothe"Normal Form 92 DIFFERENTIAL EQUATIONS where /#- \Pl-~\P\ and 8-Re*lrdm . Putinto itsNormal Form, andhence solve yz-4:xy l+(4:X2-!)?/=-3ex*sin2x. (11)Show that ifthetwoequations and reduce tothesame Normal Form, theymay betransformed into each other bytherelation i.e.thecondition ofequivalenceisthat theInvariant Ishould bethe same. (12)Show thattheequations and x have thesame invariant, andfindtherelation thattransforms oneinto theother.Verify byactually carrying outthistransformation. (13)Ifuandsuareanytwosolutions of t>,+It>=0,.................................... (1) prove that ??=,-2^, .................................... (2) andhence that From(2)show that ifsisanysolution of(3),s^andss^are solutions of(1). [The function ofthedifferential coefficients ofsontheleft-hand side of(3)iscalled theSchwarzian Derivative (after II.A.Schwarz of Berlin) andwritten{s,x}.Itisofimportanceinthetheoryofthe Hyper geometric Series.] (14)Calculate theInvariant /oftheequation Takingsasthequotientofthetwosolutions xexand x,verify that {*,a}-27, andthat$iandss{~* aresolutions oftheNormal Form oftheoriginal equation. (15)Ifuandvaretwosolutions of provethat uv2-vu2+P(uv 1-vuj=0, andhence that uv1-vul=^ae^J Verifythisfortheequationofthelastexample. MISCELLANEOUS EXAMPLES 93 (16)Show thatyyt**const, isafirstintegraloftheequation formed byomittingthelastterm of y Byputting yy1=C,whereCisnowafunction ofx(infact, varying theparameter C),show that ifyisasolution ofthefullequation, then Ci--y2 , andhence C2=const.-Jy4 , giving finally y2=asin(x<\/2+b). [Thismethodappliestoanyequationoftheform (17)Solve thefollowing equations bychanging theindependent variable : (18)Transform thedifferentialequation T-4cosx+-~-sinx-2?ycos3a;=2cos5xax2ax Intoonehavingzasindependent variable, where z=sin x, andsolve theequation. [London.] (19)Show that ifzsatisfies bychangingtheindependentvariable from a;toz,weshall transform intoa Hence solve-7-^+f1-- )T^+4x2ve~2a!=4(z2-fx3 )e~3a> .dx2\x/dxy v x P.D.B. CHAPTER VIII NUMERICAL APPROXIMATIONS TOTHESOLUTION OF DIFFERENTIAL EQUATIONS 82.Thestudent willhave noticed thatthemethodsgiveninthe preceding chaptersforobtainingsolutions infinite form onlyapply tocertainspecial typesofdifferentialequations.Ifanequation doesnotbelongtooneofthesespecial types, wehave touseapproxi- mate methods. Thegraphical method ofDr.Brodetsky, givenin Chapter I.,givesagood generalidea ofthenature ofthesolution, but itcannot berelied uponfornumerical values. Inthischapter weshall firstgivePicard's*method forgetting successivealgebraic approximations. Byputting numbers inthese, wegenerally getexcellent numerical results.Unfortunatelythe method canonlybeappliedtoalimited class ofequations,inwhich thesuccessiveintegrations canbeeasily performed. Thesecond method, which isentirely numerical and ofmuch moregeneral application,isduetoRunge.f Withproper pre- cautions itgives goodresults inmost cases, although occasionally itmayinvolve avery largeamount ofarithmetical calculation. We shall treat severalexamples bybothmethods toenable their merits tobecompared. Variations ofRange's method havebeengivenbyHeun, Kutta, andthepresentwriter. 83.Picard's method ofintegrating successive approximations. The differentialequation fa *B.Picard, Professor attheUniversityofParis, isoneofthemostdistinguished mathematicians ofto-day. He iswellknown forhisresearches ontheTheory of Functions, andhisTraiti <fanalyseisastandard text-book. f0.Runge, Professor attheUniversityofGottingen, wasanauthority on graphical methods. 94 NUMERICAL APPROXIMATIONS 95 where y=6when xa,canbewritten y=6+ Jf(x 9y)dx. Forafirstapproximation wereplacetheyin/(x, y)by6;for asecond wereplaceitbythe firstapproximation,forathirdbythe second, andsoon. Ex.(i). ~=*x+1/1 ,where t/=0when x=0. Here y=I(x+y*)dx. Jo Firstapproximation.Putt/=inx+y2 ,giving y= xdx=*%x*. Jo Second approximation. Put?/=|x2ina;+y2 ,giving t/=[ (z+i-^)<fa=z2+^zB . Jo Thirdapproximation. Put/=\xz+^a;5inx4-y2 ,giving y-f(x- Jo andsoonindefinitely. I^U Ex.fii).\fb where^=="1andz=*\whenx~0. Here t/^l+lzdxand i2!s=ii-l'l Jo Jo First approximation. yl+f Jo 2|+fJo Second approximation. 96 DIFFERENTIAL EQUATIONS Thirdapproximation. andsoon. Ex.(iii). l=*3 (-|+ y)>w^rey=land^-Jwhens0. Byputting -^=z,wereduce thistoEx.(ii).ax Itmay beremarked that Picard's method converts thedifferential equationintoanequation involving integrals, which iscalled anIntegral Equation. Examples forsolution. Find thethird approximation inthefollowingcases. Forexamples (1)and(2)obtain alsotheexact solution bytheusual methods. (1)-^=2t/-2o;2-3,wherey2when z=0.dx (2)-|[-2--,wherey=2when a=l. (3) where y=2and2=0when (4) wheret/=5andz=*lwhen x ._.d2y dy * * <,^V * xv (5)-j-2=z -JT-f-#*y,where v=5andV-^l when 05=0.ax* ax ax 84.Determination ofnumerical values from these approximations. Supposethat inEx.(i)ofthelast article wedesire thevalue ofy, correct tosevenplacesofdecimals, when x=0-3. Substituting x=0-3,weget (0-3)2=0-045 from thefirstapproxi- mation. Thesecond addsA(0'3)5-0-0001215, while thethird adds T^r(0-3)8+TJibTr(0-3J11=0-00000041 .... NUMERICAL APPROXIMATIONS 97 Noticingtherapidwayinwhich these successive increments decrease, weconclude that thenext onewillnot affect the first seven decimalplaces,sotherequiredvalue is0-0451219... . Ofcourse forlargervalues ofxweshould have totakemore than threeapproximationstogettheresult totherequired degree ofaccuracy. WeshallproveinChap.X.thatunder certain conditions the approximationsobtainedreally dotend toalimit,andthat thislimit givesthesolution. This iscalled anExistence Theorem. Example forsolution. (i)Show that inEx.(ii)ofArt. 83,x=0-5gives y=1-252... and z=0-526...,whilez=0-2gives y=M00025. ..and2=0-500632... . 85.Numerical approximation direct from thedifferential equation. Themethod ofintegratingsuccessiveapproximationsbreaks down if,asisoften thecase, theintegrationsareimpracticable.But there areother methods which canalwaysbeapplied.Consider theproblem geometrically. The differential equation j-/*> determines afamilyofcurves(the"characteristics ")which donot intersect each other and ofwhich onepasses through every point FlO. 23. intheplane.*Given apointP(a,6),weknow thatthegradient ofthecharacteristic through Pis/(a, 6),andwewant todetermine *This isontheassumptionthatf(x, y)hasaperfectlydefinite value forevery pointintheplane. If,however, f(x, y)becomes indeterminate foroneormore points, these pointsarecalled singular pointsoftheequation, andthebehaviour ofthecharacteristics nearsuch pointscalls forspecial investigation. SeeArt. 10t 98 DIFFERENTIAL EQUATIONS they**NQ ofanyotherpointonthesame characteristic, giventhat x**ON a+h, say.Afirstapproximationisgiven bytakingthe tangent PRinstead ofthecharacteristic PQ,i.e.taking y~NL+LR-NL +PL tan/_RPL 6+hf(a, b)=6+A/,say. But unless hisverysmall indeed, theerrorRQisfarfrom negligible. Amore reasonable approximationistotake thechordPQas paralleltothetangenttothecharacteristic through S,themiddle pointofPR. Since5is(a-t-|A, 6+|A/ ),thisgives Thissimpleformulagives goodresults insome cases, aswillbe seenfrom thefollowing examples: Ex.(i)~=z+t/2 ;giventhaty=0when =0,required ywhen x-0-3.ax Here a=&=0, A=0-3,f(x,y)~x +y*. Therefore giving,6+A/(a+JA,6+P/)=0+0-3x/(0-15, 0)==0-045. Thevalue found inArt.84was 0-0451219...,sotheerror is 0-00012...,about|percent. Ex.(ii).~2-- :giventhatt/=2when x=1,findywhen x=1-2. tfcc x Here a=l,6-2, A=0-2, /=2-f=0. Therefore 6-fhf(a+JA,6+JA/ )=2+0-2 x/(l -1,2) -2+0-2x^2- Now thedifferential equationiseasily integrable, giving yx+-,x sowhen x-1-2 thevalue ofyis2-033... .Theerror is0-003...,which isratherlargecomparedwith theincrement ofytnamely0-036... . Ex.(iii). x,y,z),say; giventhaty=land z=>0*5whenz=0, findyand when o;=0'5. Herea=0, 6=1,c(theinitial value ofz)=0-5, A=0*5. Hence /.-/(O, 1,0-5)-0-5;g=g(0t1,0-5)0. NUMERICAL APPROXIMATIONS 99 Byanobvious extension ofthemethod fortwovariables, wetake y~b+hf(a+$h,b+J&/,c+J^7 )=1+0-5 x/(0-25, 1-125, 0-5)=1-2500, and *-c+hg(a+\h,l+Wo>c+&gQ) -0-5+0-5x#(0-25, 1-125, 0-5)-0-5127. Theaccurate values, found asinArt. 84,are y-1-252... and 2=0-526.... Thuswehave obtained afairly goodresult fory,butaverybad oneforz. Theuncertainty about thedegreeofaccuracyoftheresult deprives themethod ofmost ofitsvalue. However,itforms anintroduction to themore elaborate method ofRunge,tobeexplainedinthenext article. Examples forsolution. (1)-=,(x*-y)1;giventhaty=4when x=2-3,obtain thevaluedx f/=*4-122 when x=2-7. [Runge's methodgives 4-118.] (2)-r=*T^{y -l+log e(x+y)};giventhaty=2when x-1,obtain thevaluet/=2-194when x1.[Runge's methodgives 2-192.] (3)"=2$-- ;giventhaty=2when =1,obtain thevalue?/=2-076 eto x 24when sc=l-2. Alsoshow thaty^^x2*,sothatwhen x=l'2, t/is really2-071... .6 6x 86.Runge's method. Supposethatthefunction ofydefined*by ^"/(^y)'y~bwhen^"^ isdenoted byy=F(x). Ifthiscanbeexpanded byTaylor's theorem, NowW=-/(*,?)=/> say. Weshallnowtake the total differential coefficient withrespect tox(that is,takingtheyin/tovaryinconsequenceofthevariation ofx).Letusdenotepartialdifferential coefficients by y dfa2/a2/.a2/.p=>- 9q**^* r=3 >5s=5a>^ai? Fdx*dydx23xdy dy* and their values when x**aandy=*bby j? ,},etc. *Theconditions under which thedifferential equation andthe initial con- ditionreally dodefine afunction arediscussed inChap. X.Thegraphicaltreat- ment ofthelastarticle assumes thatthese conditions aresatisfied. 100 DIFFERENTIAL EQUATIONS Similarly,*-"<*)-(;?+|i)<P+/5) Thus-r+pq +fs+ ^ .(1) The firsttermrepresentsthe firstapproximationmentioned and rejectedinArt. 85. Thesecond approximationofArt. 85, i.e. y-b=hf(a +\h,&+Wo)~&i> say, maynowbeexpanded andcomparedwith(1). Now, byTaylor'stheorem fortwoindependent variables, -/o giving *!-A/o+I*1(p+/cflo)+i*8 fro+2/o a+/%)+.......... (2) Itisobvious that&tisatfault inthecoefficient ofA8 . Ournextstepissuggested bytheusual methods*forthe numericalintegrationofthesimplerdifferentialequation Oursecondapproximationinthiscasereduces totheTrapezoidal Ruley~&=A/(a-4A). Now thenextapproximationdiscussed isgenerally Simpson's Rule, whichmaybewritten Ifweexpandthecorrespondingformula intwovariables, namely JA{/+4/(a+i weeasilyobtain /S)+...,........ (3) which isabetterapproximationthan kltbutevennowhasnotthe coefficient ofA3quiteinagreementwith(1). Toobtain theextra terms inA3 ,Runge freplaces *Seethetext-books onCalculus byGibson orLamb. tMathematiache AnnaUn, Vol.XLV1.pp.167-178. NUMERICAL APPROXIMATIONS 101 byV'9-hf(a+h,b+V),where ft*=hf(a+A,6+A/).Themodified formula maybebrieflywrittenJ(ft'+44!4-ft'"}, where ft'= /i/o,or Iftj+lfts-fti+K^-^i)* whereft^Kft'+ft'")- Thestudent willeasily verifythat theexpansionofRunge's formulaagreeswith theright-handside of(1)asfarastheterms inh,h2 ,andA3areconcerned. Ofcourse thismethod willgivebadresults iftheseries(1)con- verges slowly. If/>lnumerically, werewrite ourequation andnowFQ<1numerically, andwetakeyastheindependent variable. 87.Method ofsolving examples byRnnge's rule. Toavoid confusion, thecalculations should beformed insome definite order, such asthefollowing: Calculatesuccessivelyft'hfQ, ,6+ft"). andfinallyftftj4-1(kz-ftx). Moreover, as ftxisitselfanapproximationtothevaluerequired, itisclear that ifthedifference between ftandftj,namely 3(&2~^i)j issmallcomparedwith ftxandft,theerror in ftislikelytobeeven smaller. Ex.(i).-^aJ-fy8 ;giventhaty=whence ==0,findywhen x==0-3. C133 Here a-0, 6-0, />=0-3, f(x,y)~x +y*t/=0; A,6+ft')-0-3x/(0-3,0) =0-3x0-3 -0-0900; U"-hf(a +h,64-i")-0-3x/(0-3, 0-09)==0-3x(0-3 -f0-0081)0-0924; tlB-A/(a+JA,6+p')=0-3x/(0-15, 0)=0-3x0-15 0-0450; ftai(ft'+ ft'")-ix0-0924 -0-0462; and -0-0454. Asthedifference between k==0-0454 and&!=0-0450 isfairlysmall comparedwith either, itishighly probablethat theerror in ftisleas 102 DIFFERENTIAL EQUATIONS than thisdifference 0-0004. That istosay,weconclude thatthevalue is0-045, correct tothethird placeofdecimals. Wecan test thisconclusion bycomparingtheresult obtained in Art.84,viz.0-0451219... . Ex.(ii).=*-- ;giventhatt/=lwhen sc=0,findywhen cc=l. dxy~\~x This isanexample giveninRunge's original paper.Divide the rangeintothreeparts,to0-2,0-2to0-5,0-5to1.Wetakeasmall increment forthe firststepbecause/ (x,y)islargestatthebeginning. Firststep. a=0, 6-1, A=0-2, /=1; fc'=A/ -0-200; k"=hf(a+h,6+&')=0-2x/(0-2, 1-2) -0-143; 2,M43)=0-140 ; -l,1-1)=0-167; *8=J(*'+t'")-ix 0-340 -0-170; and i-ij+lttj-iJ-OaGT+O-OOl -0-168 ; giving y=1-168 when x=0-2, Secondstep. a=0-2, 6=1-168, A=0-3, /=/(0-2,1-168)=0-708. Proceedingasbefore weget^=0-170, &2=0-173 andso&=0-171, giving*y-l-168-f0-171=1-339 when a=0-5. Thirdstep. a=0-5, 6=1-339, A=0-5. Wefind^=2=k=0-160, giving t/=l-499 when a=l. Consideringthekandkjttheerror inthisresult should belessthan 0-001 oneach ofthe firstandsecondstepsandnegligible (to3decimal places)onthethird, thatis,lessthan0-002altogether. Asamatter offact, thetruevalue ofyisbetween 1-498and 1-499, sotheerror islessthan 0-001. Thisvalue ofyisfound byintegrating theequation, leadingto ir-2tan-1-=log,(x2+y2 ).x Examples forsolution. Givenumerical results tothefollowing examplestoasmany places ofdecimals asarelikelytobeaccurate : (1) 7~{*/--l+log (z+t/)};giventhatf/=2wheng--1,find ywhen x=l,takingA=2(as/isvery small). (2)Obtain acloserapproximation tothepreceding question by taking twosteps. (3)-^=(xa-y)*-l ;given thatt/=4when x=2-3,findywhen jc-2-7(a)inonestep, (6)intwosteps. NUMERICAL APPROXIMATIONS 103 (4)Show that if~^=2~-andt/=2when x=*l, then y=ce+-.dx x x Hence findtheerrors intheresultgivenbyRunge's method, taking (a)&0-4, (6)h0-2, (c)^=0-1 (asingle stepineach case), andcompare these errors with their estimated upperlimits. (5)IfE(h)istheerror oftheresult ofsolving adifferentialequation ofthe firstorderbyRunge's method, provethat Hence show thattheerror inatwo-stepsolution should beabout |-ofthatgiven byonestep;that istosay,wegettheanswer correct toanextra placeofdecimals (roughly) bydoubling thenumber ofsteps. 88.Extension*tosimultaneous equations. Themethod iseasily extended tosimultaneousequations. Astheproofisverysimilar tothework inArt. 86,thoughratherlengthy, weshallmerely give anexample.Thisexample andthosegivenforsolution "aretaken, withslight modifications, fromRunge's paper. Ex.2-2*-|-/(*,y,z),8ay. giventhaty=0-2027 and 3=1-0202 when x=0-2, findyand zwhen x=0-4. Here a=0-2, 6-0-2027, c-1-0202, /=/(0-2, 0-2027, 1-0202)-1-027, Jk'=,A/ =:0-2x1-027 -0-2054; Z'=%=0-2x0-2070 -0-0414; htb+k',c+1')=0-2 x/(0-4, 0-4081, 1-0616) -0-2206; b+k',c+l')=0-2x0(0-4, 0-4081, 1-0616) -0-0894; 6-f4",c-fH=0-2 x/(0-4, 0-4233, 1-1096) =0-2322; b+k",+0-0-2x0(0-4, 0-4233, 1-1096) ~0-0934; 6-fP',c+JO=0'2x/(0-3, 0-3054, 1-0409)=0-2128; 3,0-3054, 1-0409) -0-0641; -0-2188; -0-0674; -0-2148; +0-0011 -0-0652; giving y-0-2027 +0-2148 =0-4175 and 2=1-0202+0-0652 -1-0854, probablycorrect tothethirdplaceofdecimals. *Therestofthischapter maybeomitted onafirstreading. 104 DIFFERENTIAL EQUATIONS Examples forsolution. (1)With theequationofArt. 88,show that ify-04175 and zl-0854 when x=0-4,then?/=0-6614 and=1-2145 (probablycorrect tothethirdplaceofdecimals) when #=0-6. .-0.7,00 and r=0-6when 2=1-2145, obtain thevaluesw=0-5163 andr=0-7348 when z(whichistobetaken astheindependent variable)=1-3745. Show thatthevalue ofrisprobablycorrect tofourdecimalplaces,but thatthethirdplaceinthevalue ofwmaybeinerror. (3)Byputting10=cos<f>inthelastexample andt/=sin0,xrin theexampleofArt. 88,obtain ineach casetheequations dzt_sin <t> dd> 3-=tan d>;2z=-r-+cos<f>~,dr^r^dr whichgivetheform ofadropofwaterrestingonahorizontalplane. 89.Methods* ofHeun andKutta. These methods arevery similar tothose ofRunge,soweshall statethem very briefly. The problemis :given that -T^=/(#, y)andy=bwhen x=a, tofind theincrement kofywhen theincrement ofxish. Heun calculatessuccessively f-VM), 4"-*/(*+**,&+**'). andthen takesJ(jfe'+3i'")astheapproximatevalue ofk. Kutta calculatessuccessively, andthen takes%(k'+$k"+3kf"+k"")astheapproximatevalue oft. Theapproximations canbeverified byexpansioninaTaylor's series, asinRunge'scase. Example forsolution. Given that-r^^ andv^lwhenx0, findthevalue ofy(to8dxy+xy y significant figures) whenaj=02 bythemethods ofRunge, Heun, and Kutta, andcompare them with theaccurate value 1-1678417. [From Kutta'spaper. ] *Ztfochrifl fiirMatfomeiik undPhysik, Vols. 45and46. NUMERICAL APPROXIMATIONS 105 90.Another method, with limits fortheerror. Thepresentwriter hasfound*fourformulae whichgivefournumbers, between the greatest and least ofwhich therequiredincrement ofymust lie. Anewapproximate formula canbederived from these. When appliedtoRunge's example,thisnewformulagivesmore accurate results thananyprevious method. Themethod isanextension ofthefollowingwell-known results concerningdefiniteintegrals. 91.Limits between which thevalue ofadefinite integral lies. Let F(x) beafunction which, togetherwith its firstandsecond differential coefficients,iscontinuous (and thereforefinite) between x=aandx=a+h.LetF"(x)beofconstantsignintheinterval. Inthefigurethissignistaken aspositive, makingthecurve concave upwards. LP,MQ,NRareparalleltotheaxis ofy,Misthe middlepointofLN,andSQTisthetangentatQ.OL=a, M FIG. 24. Then theareaPLNR liesbetween that ofthetrapezium SLNT andthesum oftheareas ofthetrapezia PLMQ, QMNR. a+h Thatis,1F(x)dxliesbetween Jo. say, and h{F(a)+(2F(a+%h)+F(a +h)}=B, say. Inthefigure F*(x)ispositive andAisthelower limit,Bthe upper.IfF"(x) werenegative, Awould betheupperlimitandB thelower. Phil Mag., June 1919. Most ofthispaperisreproducedhere. 106 DIFFERENTIAL EQUATIONS Asanapproximationtothevalue oftheintegralitisbest to take, notthearithmetic mean ofAandjB,but%B+$A, which is exactwhenPQRisanarcofaparabolawith itsaxisparalleltothe axis ofy.Itisalsoexact forthemoregeneralcasewhen F(x)=a+bx+cxa+ex*, asisprovedinmost treatises ontheCalculus intheir discussion of Simpson'sRule. 92.Extension ofpreceding results tofunctions defined bydifferential equations. Consider thefunction defined by -jx-/fa,y\y=bwtenx=*<*; where/(a?, y)issubjecttothefollowinglimitations intherangeof values atoa+hforxandb-htob+hfory.Itwillbeseenfrom what follows below thattheincrement ofyisnumericallylessthan h, sothat allvalues ofywill fallintheaboverange. The limitations are : (1)f(x, y)isfiniteandcontinuous, asarealso itsfirstandsecond partialdifferential coefficients. (2)Itnever numerically exceedsunity.Ifthiscondition isnot satisfied, wecangenerally getanewequationinwhich itissatisfied bytaking yinstead ofxastheindependentvariable. (3)NeitherdPy/dx?nordf/dychanges sign. LetmandMbeanytwonumbers, such that Then ifthevalues ofywhen xisa-hJAanda+haredenoted by b+jandb+krespectively,* -ift^imft<j<PfA^iA,........................(1) and -h^mh<Jc<Mh^h .........................(2) Weshallnowapplytheformulae ofthelast article, taking yto bethesame function asthatdefined by Ca \JaF(x)dx, fa+hsothat k**\ F(x)dx.Ja Wehave toexpresstheformulae interms of/instead ofF. Now, -F(a)=-the value ofdy/dxwhen z=a, sothat F(a)-f(a,b). *Thefollowing inequalitiesholdonlyifhispositive.Ifhisnegative, they must bemodified, butthefinal result stated attheendofthis article isstill true. NUMERICAL APPROXIMATIONS 107 Similarly, F(a+\h)=/(a+\h,b+j), and F(a+h)=*f(a+h, b+k). Now,ifdf/dyispositive,sothat/increases withy,theinequalities (1)and(2)leadto f(a+%h,b+%mh)<f(a+%h,b+j)<f(a+%h,b+$Mh),.....(3) and/(a+ft,b+mh)<f(a+h,b+k)<f(a +h,b+Mh);........... (4) while ifdfldijisnegative, /(a+JA,&+iro*)>/(a+|A,6-fj)>/(a +J*,6+pf*), ...(5) and/(a+A,b+mh)>f(a +h,6-fk)>f(a +h,b+Mh)............(6) Thus ifF/r (#)=(Py/dy?ispositive and9//9yisalsopositive,the result ofArt. 91, may tysreplaced by p<&<Q,..................................(7) where p=hf(a+JA,6+JwA) andQ=JA{/(a, 6)+2/(a +JA,6+JMA) -f/(a+A,6+MA)} ; while ifJ"(x)ispositive, anddf/dyisnegative, P<k<q,...................................(8) where P=hf(a+\h,b+JMA) and?=4A{/(, 6)+2/(a+JA, 6+mA)+/(a +A,6+m/i)}. Similarly,if^(x) and9//3yarebothnegative, p>k>Q,..................................(9) while ifF"(x)isnegativeand9//9y positive, P>k>q..................................(10) These results maybesummed upbysayingthat ineverycase (subjecttothelimitations on/stated atthebeginningofthisarticle) kliesbetween thegreatestand leastofthefournumbersp,P,q,andQ. Asanapproximateformula weuse&==B+^A, replacing Bby Qorq,andAbyporP. 93.Application toanumerical example. Consider theexample selected byRunge andKutta toillustrate their methods, dyy-x , ,-.J^z-t/!when z=0.dxy+35y Itisrequiredtofindtheincrement kofywhen xincreases by 0-2. Here/(x, y)=(y-x)/(y+x).This function satisfies thecon- ditions laiddown inthelastarticle.* WetakeM=l, ro=(l-0-2)/(l-2+0-2)=4/7. *Asf(x,y)ispositive, yliesbetween 1and 1-2.When findingMandmwe always takethesmallest rangeforythatwecanfind. (The conditionsm<f<M canbereplaced bym^f^M, without affecting thefinal result excepttoreplace some<signsby^signs.) 108 DIFFERENTIAL EQUATIONS Then p-0-1654321, P=0-1666667, q-0-1674987, Q=0-1690476. Thus Icliesbetween pandQ.Errors. lQ+lp-0-1678424, 0-0000007 Kutta's value 0-1678449, 0-0000032 Runge'svalue 0-1678487, 0-0000070 Heun's value 0-1680250, 0-0001833 Thesecond, third, andfourth ofthese were calculated byKutta. Now thisparticular exampleadmits ofintegrationinfinite terms, giving log(x2+*/2 )-2tan-1(x/y)=0. Hence wemayfindtheaccurate value ofk. Accurate value =0-1678417. Thus inthisexampleourresult isthenearest totheaccurate value, theerrorsbeingasstated above. Wemayalso testthemethod bytakingalargerinterval Al. Ofcourse amore accurate wayofobtainingtheresult would beto take severalsteps, sayh=0-2, 0-3,andfinally 0-5,asRungedoes. Still, itisinterestingtoseehow farwrongtheresults come for thelargerinterval. Wetake M-l, w=(1-l)/(2 +1)=0. Then |Q4-^^=0-50000. True value-0-49828, Errors. Kutta's value =0-49914, 0-00086 Ourvalue =0-50000, 0-00172 Heun's value =0-51613, 0-01785 Runge'svalue=0-52381, 0-02553 ThistimeKutta's value isthenearest, andours issecond. [For asystematic method ofdetermining Mandm,and for Remes* extension ofthemethod ofArts. 90-93, seeArt. 183. .ForAdams' numerical method, perhapsthebest ofall,seeArt, 182.] CHAPTER IX SOLUTION INSERIES. METHOD OFFROBENIUS 94.InChapter VII.weobtained thesolution ofseveralequations oftheform wherePandQwere functions ofx. Inevery casethesolution wasoftheform y=af(x)+bF(x) 9 where aand6werearbitraryconstants. Thefunctions/(x)andF(x)weregenerally made upofintegra orfractionalpowersofx,sines and cosines, exponentials,an< logarithms,such as ,i i (l+2x)ex ,sinz+zcosz, x*-fx,x+logx,ex . The firstandsecond ofthese functions canbeexpanded b; Maclaurin's theorem inascending integral powersofx;theother cannot, thoughthelastcanbeexpandedinterms of1/x. Inthepresent chapter, followingF.G.Frobenius,* ofBerlin,w shallassume asatrial solution y=tf (ao+ajX-f a<p?+...toinf.), where thea'sareconstants.*)" Theindex cwillbedetermined byaquadratic equationcalle< theIndicialEquation. Theroots ofthisequation maybeequa different anddiffering byaninteger,ordifferent anddiffering by quantitynotaninteger.These cases willhave tobediscusse separately. Thespecialmerit oftheform oftrial solution usedbyFrobeniu isthat itleads atonce toanother form ofsolution, involving logc when thedifferential equationhasthissecond form ofsolution. *Crelle, Vol.LXXVL, 1873, pp.214-224. tInthischaptersuffixes willnotbeused todenote differentiation. P.D.B. 109 I 110 DIFFERENTIAL EQUATIONS 1 Assuchafunction asexcannot beexpandedinascending powers ofx,wemustexpectthemethod tofailfordifferentialequations havingsolutions ofthisnature. Amethod willbepointedoutby which canbedetermined atoncewhichequationshave solutions of Frobenius' forms(regular integrals) and forwhatrangeofvalues ofxthese solutions willbeconvergent. Theobjectofthepresent chapteristoindicate how todeal withexamples.Theformalproofsofthetheoremssuggestedwill begiveninthenextchapter. Amongtheexampleswillbefound theimportant equationsof Bessel,* Legendre,andRiccati. Asketch isalsogivenoftheHyper- geometricorGaussian equation and itstwenty-foursolutions. 95.Case I.Roots ofIndicial Equation unequal and differing "bya quantity notaninteger. Consider theequation Put z=a?(a+a1z+<z2#2+),where a^0, giving +a2(c d?zand- Substitute in(1),Jandequatethecoefficients ofthesuccessive powersofxtozero. Thelowest powerofxisof"1 .Itscoefficientequatedtozerogives o{2c(c-l)-c}-0, i.e.c(2c-3)=0,...........................(2) *Friedrich Wilholm Bessel, ofMinden (1784-1846), wasdirector oftheobser- vatoryatK.6ni<jsberg. He isbestknown by"Bessel'B Functions." Adrian Marie Legendre, ofToulouse (1752-1833),isbestknown byhis"Zonal Harmonics" or"Legendro's Coefficients." Healsodidagreat deal ofworkon Elliptic Integrals andtheTheory ofNumbers. Jacopo Francesco, Count Riccati, ofVenice (1676-1754), wrote on"Riccati's Equation," andalsoonthepossibility oflowering theorder ofagiven differential equation. Karl Friedrich Gauss, ofBrunswick (1777-1855), "the Archimedes ofthe nineteenth century," published researches onanextraordinarily wide range o/ subjects, including Theory ofNumbers, Determinants, Infinite Series, Theoryof Errors, Astronomy, Geodesy, andElectricity andMagnetism. fItislegitimatetodifferentiate aseries ofascending powersofxtermbyterm inthismanner, within theregionofconvergence. SeeBromwich, Infinite Series, Art.52. tOrrather inwhat(Hbecome* whenijisreplaced byz. SOLUTION INSERIES 111 (2)iscalled theIndidalEquation. The coefficient ofofequatedtozerogives o3{2(c+l)c-(c +l)}0,i.e.^=0................(3) The coefficient ofaf+1hasmore terms init,giving i.e.a2(2c+l)+a(0-3)=0.........................(4) Similarly,a3(2c+3)-fo^c-^J-O,........................(5) a4(2c+5)+a2(c-!)=(),........................(6) andsoon. From(3), (5),etc.,0=a1a3=ai...a2n+1. From(4), (6),etc., 2C-304 C-l Oo"~2c+l'" Butfrom(2),c=0or|. Thus,ifc=0, =aw, say, replacingaQbya;and if ^=6vsay,replacing (whichisarbitrary) by6thistime. Thusy^au +bvisasolution which contains twoarbitrarycon- stants, andsomaybeconsidered thecomplete primitive. Ingeneral, iftheIndidalEquationhastwounequalrootsaand/3 differing byaquantitynotaninteger, wegettwoindependentsolutions bysubstitutingthese valuesofcintheseriesforz. Examples forsolution. (3) (4)Bessel'sequationoforder n,taking 2nasnon-integral, 112 DIFFERENTIAL EQUATIONS 96.Convergence ofthe series obtained inthe last article. Itis provedinnearly everytreatise onHigher AlgebraorAnalysisthat theinfinite series%+u2+u3+...isconvergentif Lt<L n->o> Now intheseriesweobtained un=a2n-2a^+2n~2 >**& n-a andthelimitwhen w->o> is-Jxa ,independentofthevalue ofc. Hence both series obtained areconvergentfor |x \<<\/2. Itisinterestingtonotice that ifthedifferentialequationia reduced totheform 1 givinginourexample p(x)=~-^A~\~X p(x)andy(x)areexpansibleinpowerseries which areconvergent forvalues ofxwhose modulus |x |<\/%. Thatis,theregionofconvergenceisidentical inthisexample withtheregionforwhichp(x)andq(x)areexpansibleinconvergent powerseries. Weshallshow inChap.X.that thistheorem istrue ingeneral. Examples forsolution. Find theregionofconvergenceforthesolutions ofthelast setof examples. Veiifyineach casethattheregionofconvergenceisidentical with theregionforwhichp(x)andq(x)areexpansibleinconvergent powerseries. 97.Case II.Roots ofIndicial Equation equal Consider the equation Put z^^(aQ+a^x+a^c2+...), andaftersubstitutinginthedifferentialequation, equatecoefficients ofsuccessive powersofxtozerojustasinArt. 95. SOLUTION INSERIES 113 Wegeta{c(c-l)+c}=0, i.e.c*=0, .................................... (1) andsoon. Hencei.e.a1(c+l)2-a(c+2)2=0, ........................(2) a2(c+2)2-^(0+3)2-0,........................(3) <x2(c-f4)2 ===0,........................(4) x+\...I J isasolution ife=0. Thisgives onlyoneseries instead oftwo. But ifwesubstitute theseries intheleft-hand side ofthedif ferential equation (without putting c=0), wegetthesingleterm ac?3f~l .Asthisinvolves thesquareofc,itspartialdifferential coefficient withrespecttoc,i.e.2aQcxc'1+acV"1 logx,will also vanish when c=0. Thatis, lc[(x"xZ)^+(l~5x)fx* *]*=2<V*C-1+a^V^loga Asthe differentialoperatorsarecommutative, thismaybe written s .Hence -^-isasecond solution ofthedifferentialequation,ifcis putequaltozero after differentiation. Differentiating, Puttingc=>0anda=aandbrespectivelyinthetwo series, and =*bulogx-26{1.2x-f2.3#2-1-3 .4x8+...}<=bvtsay. Thecomplete primitiveisau+bv. 114 DIFFERENTIAL EQUATIONS Ingeneral, iftheIndidalEquationhastwoequalroots c=a, wegettwoindependentsolutionsbysubstitutingthisvalueofcinzand ~-.Thesecond solution willalwaysconsist oftheproductofthe first solution(oranumericalmultipleofit)andlogx,added to another series. Revertingtoourparticular example,consideration ofp(x) andq(x),asinArt. 96,suggeststhat the series willbeconvergent for |x |<1.Itmaybeeasily shown that this iscorrect. Examples forsolution. <i)(*-*')g+(i-)g-jr-a (2)Bessel's equationoforder zero ,4) 98.Case III.Roots ofIndicial Equation differing byaninteger, making aCoefficient ofzinfinite. Consider Bessel'sequationoforder d*y dy- IfweproceedasinArt. 95,wefind afl{c(c-l)+c-l}=0, i.e.c"-l=0, ........................(1) i.e.Oj-0, (2) !+2)2-l}+a =0, (3) It f\ IA\ giving 1 ~a#f\\- 7^(c+l)(c+3)2 (c __1__ (c+l)(c+3)*(c+6) Theroots oftheindicial equation (1)arec=1or-1. But ifweputc=-1inthis series forz,thecoefficients become infinite, owingtothefactor(c+1)inthedenominator. SOLUTION INSERIES 115 Toobviate thisdifficulty replace*aby(c+l)Jc,giving f 1 1-faM(c +l)-a?+ -.- JTT (c+3) ^+"T......( anda*j~+s+(x-^ Just asinCase II.theoccurrence ofthesquaredfactor(c-fl)a shows that ~~,aswell asz,satisfies thedifferential equation when c=-1. Alsoputtingc=linzgivesasolution. Soapparently we have found three solutions tothis differential equationofonlythe second order. Onworking them out,wegetrespectively 221 and Itisobvious thatw=>-4w,sowehaveonlyfound twolinearly independentsolutions after all,andthecomplete primitiveisau+bv. The series areeasily provedtobeconvergentforallvalues ofx. Theidentity (exceptforaconstantmultiple)oftheseries obtained bysubstitutingc=-1and c1respectivelyintheexpressionforz isnotanaccident. Itcould havebeenseenatoncefrom relation(4), Ifc-1, thisgivesan{(l+w)2-l}+a n_2=0......................(6) Ifc--l, an{(- hencereplacing nbyn+2, l}+0n=0......................(7) Thus["?*] -f^s-l .........................(8)LanJiLan_2Jc=1 As[zjc^-! hasor1asafactor outside thebracket, while[z]c=ihas ,relation(8)reallymeans that thecoefficients ofcorresponding *Ofcourse thecondition a^isthus violated;weassume initsplacethat 116 DIFFERENTIAL EQUATIONS powersofxinthetwoseries areinaconstant ratio. The first series apparently hasanextra term, namelythatinvolvingar1 ,butthis conveniently vanishes owingtothefactor(c+I). Ingeneral, iftheIndicial Equationhastworootsaand/3(say a>/3) differing byaninteger, andifsomeofthecoefficients ofzbecame infinite when c=/3,wemodifytheform ofzbyreplacingabyk(c-/8). Wethengettwoindependent solutions byputting c=*/3inthemodified form ofzand ~- .The resultofputtingcainzmerely gives a numericalmultiple ofthatobtainedbyputting c/3. Examples forsolution. (1)Bessel'sequation oforder 2, (4) 99.Case IV.Roots ofIndicial Equation differing byaninteger, making acoefficient ofzindeterminate. Consider theequation Proceedingasusual, weget c(c-l)-0,...............(1) =0, ...............(2) -0,...............(3) 0,...............(4) andsoon. (1)Givesc=0 or1. The coefficient ofalin(2)vanishes whenc=0, butasthere isno other term intheequationthismakesa^indeterminate instead of infinite. Ifc=l,%=(). Thus,ifc=0,fromequations (3), (4),etc. etc.f SOLUTION INSERIES 117 giving This contains twoarbitrary constants, soitmaybetaken asthe complete primitive. The seriesmay beproved convergentfor I*KI. Butwehave theother solutiongiven byc=l.Workingout thecoefficients, thatis,aconstantmultipleofthesecond series inthe firstsolution. Thiscould have been foreseen fromreasoningsimilar tothat in Case III. Ingeneral, iftheIndicialEquationhastworoots aand/3(say a>/3) differing byaninteger, andifoneofthecoefficients ofzbecomes indeterminate whenc=r/3,the'complete primitiveisgiven byputting c=/3 inz,which thencontains twoarbitraryconstants. The resultoj puttingc=ainzmerely givesanumericalmultiple ofoneojtheseries contained inthefirstsolution. Examples forsolution. (1)Legendre's equationoforderunity, (1_^_2a;^v 'dx* dx (2)Legendre's equationoforder n, 100.Some caseswhere themethod fails. Asexcannot beexpanded inascending powersofx,wemustexpectthemethod tofailin somewaywhen thedifferentialequationhassuchasolution. To construct anexample,take theequation -ri-y^Q,ofwhich e* (IZ' -I ande~zaresolutions, andtransform itbyputting z=-.x *TTT i dydxdt/ 1dy dy *Wehave-f=-,-- -/--- -f--x*/-dzdzdx z*dx ax , d*ydxd/dy\ 9df 2%\ .d^y dyand j-* -j-j -j)=-x2 i(-x2-f )=cc4-j\+2X8 -/-.dz2dzdx\dz/ dx\ dx/ dx2dx 118 DIFFERENTIAL EQUATIONS Hence thenewequationis _..dxy Ifwetrytoapplytheusual method, wegetfortheindicia! equation, -a=>0,which hasnoroots,* asbyhypothesisa=f:0. Such adifferential equationissaid tohave noregular integrals \ _i_ inascending powers ofx.Ofcourse exand e*canbeexpanded in .1 powersof-. Theexamples givenbelow illustrate otherpossibilities,such as theindicial equation havingoneroot,whichmayormaynotgive aconvergentseries. Itwillbenoticed that, writingtheequationintheform ineverycasewhere themethod hassucceededp(x)andq(x)have been finite foroj=0, while inallcases offailure thiscondition is violated. Forinstance, intheaboveexample, J>(s)-2, q(x)=* --g,which isinfinite ifz=0, Examples forsolution. (1)Transform Bessel'sequation bythesubstitution x=1/2. Hence show that ithasnointegralsthat areregularindescending powersofx. (2)Show that thefollowing equation hasonlyoneintegralthat is regularinascending powersofx,anddetermine it : 2z)^-2y=0.'dxy (3)Byputting y=*vx2 (l+2z)determine thecomplete primitiveof theprevious example. (4)Show thatthefollowing equationhasnointegral that isregular inascending powersofx,astheone series obtainable divergesforall values ofx: fa (5)Obtain twointegralsofthelastexample regularindescending powersofx. Orwemaysaythat ithastwo infinite roots. SOLUTION INSERIES 119 (6)Show that thefollowing equationhasnointegrals that are regular ineither ascendingordescending powersofx: [Thisistheequation whoseprimitiveis MISCELLANEOUS EXAMPLES ONCHAPTER IX. (1)Obtain threeindependentsolutions of *dxa (2)Obtain three independent solutions, oftheform dz ,d*z *>We'andW oftheequationza^+3zy^+(l-z)-~ -t/==0. (3)Show thatthetransformation yr-"r~ reduces Riccati'sequation dii -~-+oy2=cxm ax tothelinear formj^bcvx. (4)Show that ifyianeither zeronoraninteger,theHypergeometric Equation hasthesolutions(convergentif\x\<1) F(a, P,y,x)and xl-yF(a-y+1,/8~y+l,2-y, x), where F(a, /3,y,x)denotes theHypergeometricSeries , ,aft l.y 1.2.y(y +l) 1 .2 .3.y(y+l)(y+2) (5)Show that thesubstitutions #=1-zandxl/ztransform the hypergeometric equationinto and (!-2)+{(!-a-^-(2-y)2}+afy=0 respectively,ofwhich the first isalso ofhypergeometricform. 120 DIFFERENTIAL EQUATIONS Hence, from thelastexample, deduce thattheoriginal equationhas theadditional foursolutions : (l-x)r *F(y-ft y-a,1+y-a-ftl-s), x-"F(a, a+l-y, a+l-ftor1 ), and ar0F(ft /3+1-y, /3-f1-a,ar1 ). (6)Show that thesubstitution ^=(1-x)nYtransforms thehyper- geometric equationintoanotherhypergeometric equationif w=y-a-/3. Hence show that theoriginal equationhastheadditional two solutions :(i_s)Y--/jF( y_0|y_y,x) and xl-y(l-x)y-*~f*F(l-a, 1-ft2-y, a?). [Note. Ex.5showed howfrom theoriginal twosolutions ofthe hypergeometric equation twoothers could bededuced byeach ofthe transformations x=*l-z andx=l/z. Similarly each ofthethree 1 z zI transformations x=-- ,x=-,x=- ,givestwomore, thusmaking1z z J. z twelve. ByproceedingasinEx.6thenumber canbedoubled, giving atotal oftwenty-four. These fivetransformations, together with the identical transformation xz,form agroup ;thatis,byperforming two suchtransformations insuccession weshallalways getatransformation oftheoriginalset.] (7)Show that, unless 2nisanoddinteger (positive ornegative), Legendre's equation hasthesolutions, regularindescending powersofx, [The solution forthecase2n=-1canbegotbychangingxinto ar1intheresult ofEx.4ofthesetfollowingArt. 97.] (8)Show that theform ofthesolution ofBessel'sequationof orderndepends upon whether niszero, integral,ornon-integral, although thedifference oftheroots oftheindicialequationisnotn but2n. CHAPTER X EXISTENCE THEOREMS OFPICARD, CAUCHY,f AND FROBENIUS 101.Nature oftheproblem. Inthepreceding chapters wehave studied agreatmany devices forobtainingsolutions ofdifferential equationsofcertainspecialforms. Atonetime mathematicians hoped thattheywould discover amethod forexpressingthesolution ofanydifferentialequationinterms ofafinitenumber ofknown functions ortheirintegrals. When itwas realised that thiswas impossible, thequestionarose astowhether adifferentialequation ingeneral hadasolution atall,and,ifithad, ofwhat kind. There aretwo distinct methods ofdiscussingthisquestion. One,due toPicard, hasalready been illustrated byexamples (Arts. 83arid84).Weobtained successiveapproximations, whichapparently tended toalimit.Weshallnowprovethat theseapproximations reallydotend toalimit and that this limitgivesthesolution. Thusweshall provethe exist- ence ofasolution ofadifferentialequationofafairly general type.Atheorem ofthiskind iscalled anExistence Theorem. Picard's method isnot difficult, sowewillproceedtoitatonce beforesaying anythingabout thesecond method. Itmust be borne inmind that theobjectofthepresent chapterisnotto obtainpracticallyuseful solutions ofparticular equations. Our aimnow istoprovethat theassumptions made inobtaining these solutions were correct, andtostate exactlytheconditions that aresufficient toensure correctness inequationssimilar to those treated before, butgeneralisedasfaraspossible. *Thischapter should beomitted onafirstreading. tAugustn Louis Ciuchy.ofParis (1789-1857), may belooked upon asthe creator oftheTheoryofFunctions andofthemodernTheoryofDifferential Equa- tions. Hedevised themethod ofdetermining (Jefinjte integrals byContest Integration. 121 122 DIFFERENTIAL EQUATIONS 102. Picard's method ofsuccessive approximation. If-?=/($, y} andj/=6when x=a,thesuccessiveapproximationsforthevalue ofyasafunction ofxare IXf(x,b)dx=ylysay a fJC. b+\f(x yy^dx=*y& say,andsoon. Ja Wehavealready (Arts. 83and84)explainedtheapplicationof thismethod toexamples. Wetook thecasewhere/(x,y)*=x+y*z fr=a=0, andfound These functions appeartobetendingtoalimit, atanyrate for sufficientlysmall values ofx.Itisthepurposeofthepresent article toprovethat this isthecase, notmerelyinthisparticular example,,butwheneverf(x, y)obeyscertain conditions tobe specified. These conditions arethat, after suitable choice ofthepositive numbers hand k,wecanassert that, for allvalues ofxbetween a-handa+ft,and forallvalues ofybetween b-kand6-fk,we canfindpositive numbersMandAsothat (i)\f(*,y)\<M, (ii)\f(x,y)-f(x,y')\<A\y-y'\ 9yandy'being anytwo values ofyintherangeconsidered. Inourexample /(x,y)=x -fy2 ,condition(i)isobviously satisfied. takingforManypositive numbergreater than|a |+4+{|6 1 socondition(ii)isalso satisfied, taking^t=2(|6|+k). Returningtothegeneral case,weconsider thedifferences between thesuccessive approximations. f* \ Jaf(x,b)dx,bydefinition, a but \f(x, b) |<M,bycondition(i), *..<M\x-a\<Mh (1) EXISTENCE THEOREMS 123 f*f* Alsoya-yl=6+1/(x,yi)dx-b-I/(x,b)dx,bydefinition, Ja Ja f* but|/(x, 2/x)-/(x, 6) |<4 |yl-b \,bycondition(ii), <AM |x-a |,from(1), f*Bo|y 2~"yil< IAM(x a)dx.Ja,i.e.- Similarly, |yn-yn_, |<^}MAn~lhn (3) Now theinfinite series isconvergentforallvalues ofA,A,andM. Therefore theinfinite series eachterm ofwhich isequalorlessinabsolute value than thecorre- sponding term ofthepreceding,isstillmoreconvergent. That istosaythatthesequence andsoon,tends toadefinite limit, sayY(x), which iswhatwe wanted toprove. WemustnowprovethatYsatisfies thedifferentialequation. Atfirstsightthisseems obvious, but itisnotsoreally,forwe must notassume withoutproofthat Lt If(x9yn-i)dx= f(x,Ltyn-i)dx. n->ooJa Ja n-> Thestudent whounderstands theidea ofuniformconvergence willnotice that theinequalities (1), (2), (3)thatwehave used to provetheconvergenceofourseriesreally proveitsuniform con- vergencealso. If,then, f(x, y)iscontinuous, yl9y2,etc., are continuous also,andYisauniformly convergentseries ofcon- tinuous functions; thatis,Yisitself continuous,* andY-y n_i tends uniformlytozeroasnincreases. Hence, from condition(ii),f(x,Y)-f(x92/n-i)tends uniformly fcozero. *SeeBromwich'e Infinite Series, Art. 46. 124 DIFFEREkTIAL EQUATIONS From thiswededuce that {f(x, Y)-f(x, ?/_!)} tends tozero. Ja Thus thelimit oftherelation y=&+f(*>Vn-i)dx Ja is Y=b f therefore* -r-=f(x, Y),andYbwhen x=*a. Thiscompletestheproof. 103.Cauchy's method. Theorems on infinite series required. Cauchy's method istoobtain aninfinite series from thedifferential equation, andthenproveitconvergent bycomparingitwithanother infinite series. Thesecond infinite series isnotasolution ofthe equation, buttherelation between itscoefficients issimplerthan thatbetween those oftheoriginalseries. Our firstexampleofthis method willbeforthesimplecase ofthelinearequationofthe first order dii,. dxpW'y' Ofcourse thisequation canbesolved atoncebyseparationol thevariables, giving However, wegivethediscussion byinfinite series because itis almostexactlysimilar totheslightly more difficult discussion of d*y /xdy - andotherequationsofhigherorder. Weshallneed thefollowing theoremsrelatingtopowerseries. Thevariable xissupposedtobecomplex. Forbrevity weshall denote absolute values bycapital letters, e.g.Anfor\an\. 00 (A)Apowerseries^anxnisabsolutely convergentatall o pointswithin itscircle ofconvergence Jx \R. (B)TheradiusRofthis circle isgiven by 1_T4-^n+l 7?~ ~A~~* n ->--n providedthat thislimit exists. *When differentiatingtheintegral, thestudent should remember that the integralvariessolelyinconsequenceofthevariation ofitaupperlimit. EXISTENCE THEOREMS 125 ~l ,within |x \-R. (D)Ifwehave twopower series, then forpointswithin the circle that iscommon totheir circles ofconvergence, (2?)If^an#n=*2^nX"^ora^va^ues fxwithin the circle |#|=JB, thenan=bn. (F)An<MR~n ,whereMexceeds theabsolute value ofthe sum oftheseries atpoints onacircle |x|=J? onwhich theseries isconvergent. Proofs ofthese theorems willbefound inBromwich'sInfinite Series : AinArt.82 [Art. 84in2nded.], Bisanobvious deduction fromD'Alembert's ratio test, Art. 12, CinArt.52 [Art.12becomes Art. 12-2 in2nded.], D 54, E 52, F 82 [Art. 84in2nded.]. Twotheorems onuniform convergencewillberequiredlater on, Outwewilldefer these untiltheyareneeded. dy 104.*Convergence ofthesolution inseries of =3yp(x). Letox p(x)becapableofexpansioninapowerseries2Pnx"which is o convergent everywherewithin andonthecircle |x \JZ.Weshall 00 provethat asolution y^anx"can^eobtained which is convergentwithin this circle. Substitutinginthedifferentialequation,weobtain Vnajr*-2ax"SPx"(Theorem C) 00 **(anPo+an~iPi+n-zPa++aoPn)n .(Theorem D) Equatingthecoefficients ofa?*-1 , (Theorem E) n^1.............(1) Revise Art.7before reading thefollowing. P.D.I. K [26 DIFFERENTIAL EQUATIONS Hence fortheabsolute values ofthea'sandp's, denoted bythe corresponding capital letters, weget nAn^An_lPQ+An^P1+An_3P2+...+APn_1..........(2) LetMbeapositive numberexceedingtheabsolute value of 5(x)onthecircle |x \=22, ihen Pn<MR-;.........(3) (Theorem F) therefore, from(2)and(3), DefineBn(n>0)astheright-handside of(4),and define 5asanypositive numbergreaterthanA;thenAn<Bn.MBut (A^+An_2R~i+An_3R~*+... Hence, defining Bnasabove, B-MA+(n- *.-n-^.-i+-- n A whence, dividing byBn_andusingkfor-~,sothat^A<1, thence LtMk 1_1 n"+ fiwB 7? -I n Therefore the seriesVBnxnisconvergentwithin the circle x\=R. ^(Theorem B.) 00 Stillmore therefore istheseries2anxnconvergent within the o ame circle, since An<Bn. The coefficients alfa2,...can allbefound from(1)interms of hey's,which aresupposed known, andthearbitrary constant a . 105.Remarks onthis proof. Thestudent willprobablyhave ound thelast article verydifficult tofollow. Itisimportant not ogetconfused bythedetails ofthework. Themainpointisthis. A 1eshould liketoprove that Lt^-==-.Unfortunatelythe n->oo^n~l ^ elationdefiningtheA'B israthercomplicated. We firstsimplify tbygettingridofthenquantities P,Plf...P^_ l9Still the EXISTENCE THEOREMS 12? relation istoocomplicated,asitinvolves nA's.Weneed asimple relationinvolving onlytwo.Bytakingasuitable definition ofBnwegetsuch arelation between BnandBn^19leadingto LtB-l*A~D-- T>' ^ao#n~l K Werepeatthat theobjecto*fgivingsuch acomplicateddis- cussion ofaverysimple equationistoprovideamodel which the student canimitate inother cases. Examples forsolution. (1)Prove that,ifp(x)andq(x)canbeexpandedinpowerseries convergentatallpointswithin andonthecircleX=R,then apower series convergent within thesame circle canbefound interms ofthe firsttwo coefficients (thearbitrary constants) tosatisfy [Here Hence,ifMisanynumber exceeding theabsolute values ofboth p(x)andq(x)atallpoints onthecircleX=Ry M<~(l Define theright-handsideofthisinequalityasBnandthenproceed asbefore.] (2)Prove similar results fortheequation 106. Frobenius' method. Preliminary discussion. When the student hasmastered the last article, he will bereadyfor themore difficultproblemofinvestigatingtheconvergenceof the series given bythemethod ofFrobenius. Inthepreceding chapter (which should bethoroughly known before proceeding further), wesaw that insome cases weobtained two series involving only powersofx,while inotherslogarithmswere present. Theprocedureinthe firstcase isverysimilar tothat ofthelast article. Butinthesecond caseanewdifficultyarises. The series withlogarithmswere obtained bydifferentiatingseries with 128 DIFFERENTIAL EQUATIONS respecttoaparameterc.Now differentiation isaprocessoftaking alimitandthesummation ofaninfinite series isanotherprocess oftakingalimit. Itisbynomeans obvious that theresult will bethesame whichever ofthese twoprocessesisperformed first, even iftheseries ofdifferential coefficients beconvergent. However, weshallprovethat inourcasethedifferentiation is legitimate,butthisproofthatourseriessatisfyconditions sufficient tojustify term-by-termdifferentiation isratherlongandbewildering. Toappreciatethefollowing work thestudent should atfirst ignoreallthedetails ofthealgebra, concentratinghisattention on thegeneraltrend oftheargument. When thishasbeengrasped, hecangobackandverifythelessimportant stepstaken forgranted onafirstreading. 107.Obtaining the coefficients inFrobenius' serieswhen theroots oftheindicial equation donotdiffer byaninteger orzero. Consider theexpression where p(x)andq(x)arebothexpansibleinpowerseries^Pr$* * o and2?n#nwhich areconvergentwithin andonthecircle |x |=fi. a Wearetryingtoobtain asolution ofthedifferentialequation Ifyisreplaced byoff]anx(with a=0),$(x,y,-, becomesu oo ^anaf+n {(c+n)(c+n-1)-(c+n)p(x)-q(x)} oo=S9ntf+n ,say, o where g=a{c(c-1)-pQc-qQ} and ?n=n{(c+n)(c+n-l)-j) (c4-n)-? } -n-i (Pi(c+n-1)+q l}-an_2{p2(c+n-2)+ ...-aQ(pnc+q n). Forbrevity,denote c(c-l)-y c-y by/(c), sothat(c+n)(c+n-l)-p (c+n)-g =/(c-ftt), EXISTENCE THEOREMS 129 Thenn~0if +an_a{p2(c+n-2)+&} +...+a(W+gn)..........(2) Ifwecanchoose thea'ssothat allthe#'svanish, and ifthe 00 Beries^Jan^n8obtained isconvergent,asolution of(1)willhave been obtained. Now asa^0,g^^O gives c(c-l)- Poc-q Q=Q............................ (3) This isaquadratic equationinc,and iscalled theIndicial Equation. Let itsroots beaandft. Ifeither ofthese values issubstituted forcintheequations ?i"O*ffi^O, ffs^O, >values forOj,a2,a3,...arefound intheform an-aj> n(c)l[f(6 +n)f(e+n-l) .../(c+1)],............(4) where hn(c)isapolynomialinc.Thestudent should work outthe values ofa,andaainfull ifhefindsanydifficultyatthispoint. Theprocess bywhich anisobtained from(2)involves division by/(c+n).This islegitimate onlywhen/(c+n)=f=0. Now as /(c) (c-a)(c- /3), /(c+n)=(c +n-a) (c+n-/J), BO /(a+n)=*n(a+n-/3),..............................(5) and /(/3+n)n(/3+n-a)...............................(6) Thus,ifaandftdonot differ byaninteger,thedivisors cannot vanish, sotheabove processforobtainingthea's issatis- factory.Ifa/3, onlyoneseries isobtained. 108.Convergence oftheseries soobtained. LetMbeapositive numberexceedingtheabsolute values ofp(x)andq(x)atallpoints onthecircle|a;|=B. Then P9<MR~* and Q8<MR-'9 BOthat\p8(c4-n-s)+qs\<M(C+n-s+!)#-*. From theseinequalities andfrom(2), An<M{A n_l(C+n)R-*4-...+Aa(C+l)R-}IF(c +n), ...(7) sayAn<B n,denoting theright-handside of(7)byBn.This definesBnifn>0. DefineBasany positive numbergreater thanA .This definition ofBngives Bn+iF( +*+1)-BnF(c+n)R~l~AnM(C+n -kBnM(C+n+IJfi-1 ,where ^i<1, 130 DIFFERENTIAL EQUATIONS ,Bn+l_F(c-fn)+JcM(C+n+1)sothat--- >- > ~y |+JcM(C - #|(c-f n+l)(c-fn) ~PO(C+n+1) -ry | Now forlargevalues ofntheexpression ontherightapproaches thevalue nz^ Thus Lt^-i. Therefore theseries^^n^"an(i8tiHmore theseriesVancc" o o convergeswithin thecircle\x\=R. Thus, when and/3donot differbyaninteger, wegettwo convergentinfinite seriessatisfyingthedifferentialequation. 109. Modification required when theroots oftheindicia] equation differ byzero oraninteger. When aand/3areequal, wegetone series bythismethod. When aandftdifferbyaninteger,thismethod holdsgood forthelarger one,butnotforthesmaller, for ifa-/3=r(apositive integer)/thenfrom(5)and(6) /(a+n)=n(a+n-ft)^n(n-fr), but /(ft+n)=n(/3+n-a)=n(n- r), which vanishes whenn=rygivingazero factor inthedenominator ofarwhen c=/3. AsexemplifiedinArts. 98and99ofthepreceding chapter,thismay giveeither aninfinite orindeterminate value for some ofthe a's. Thisdifficultyisremoved bymodifyingtheform assumed fory,replacingabyk(c-fi).This willmake a,al9..., r-iallzeroandar,ar+1,...allfinitewhen cisputequaltoft.This changeintheformassumed forywillnotalter therelation between theas,andsowillnotaffect theaboveinvestigationofconvergence. 110. Differentiation ofaninfinite series with respect toaparameter c,theroots oftheindicial equation differing byaninteger. InArt.107 weobtained aninfinite seriesaf^anxn ,where thea'sarefunctions o of c.Asinthepreceding chapter, wehave toconsider the differentiation ofthis series withrespecttoc,cbeing putequalto thesmaller rootftafter thedifferentiation. EXISTENCE THEOREMS 181 Now while this differentiation isbeing performedwemaycon- siderxasaconstant. The series canthen beconsidered asaseries oo offunctions ofthevariablec,say ^w(c)>where =tf+najin(c)l[f(c+ri)f(c +n-l) .../(c+1)],from(4), where a=fc(c-/3) andthefactor(c-ft)istobedivided out ifit occurs inthedenominator. Now Goursat (Cours<TAnalyse,Vol. II.2nded.*p.98)proves that if(i)allthei//sarefunctions which areanalytic and holo- morphicwithin acertainregion boundedbyaclosed contour and continuous onthiscontour, and if(ii)theseries of\]/sisuniformly convergentonthiscontour, then thedifferentiation termbyterm givesaconvergentseries whose sum isthedifferential coefficient ofthesum oftheoriginalseries. Forthedefinitions ofholomorphic andanalytic,secthebeginning ofVol. II.ofGoursat. Itwillbeseen that the\//ssatisfythese definitions andarecontinuous aslongaswekeepawayfrom values of-cthatmake them infinite. These values area-1,ft-I,a-2, $-2, etc.Toavoid these take theregioninside acircle ofcentre c=/3andofanyradius lessthanunity. We shallnowprovethat the series isuniformly convergent everywhereinside thisregion.This willproveitisuniformly convergentonthecontour ofasimilar butslightlysmallerregion inside the first. Let5beapositive integer exceedingthelargestvalue ofCwithin thelarger region. Then forallvalues ofcwithin thisregion,forvalues ofnexceed- ing 5, F(c+n)=\(c+ri)(c +n-1)-PQ(C+U)-<? |>bydefinition ofF, g ,as\u-v\> \u\- \v\, ,asP<MandQ<M, >nz+In+J,say,where 7andJareindependento n,x.orc........................................(8) Forsufficiently greatvalues ofn,sayn>w,thelastexpression tsalways positive. Let//denote themaximum value of M[Am^(C+m)R~l+Am^(C+m- 1)R~*+...+A(C+!)#m ](9) forallthevalues ofcintheregion. *p.96in4thed* 132 DIFFERENTIAL EQUATIONS Then ifEmbeanypositive numbergreater than J5m,and, if, forvalues ofn>m,Enbedefined by /rf <f/j I9-4-Vt IJ\~*-4- r?(Q-I-VYL I1^Tr'^fl'f^W \J.//7?~*W"f^W |jtZIUL i^XVj1\vi'wI** i *-'m\"'"* '*)"'J'****'/T/\\ nn2+/n+<7*(1U) sothat Bm+1^: which hasanumeratorgreaterthanandadenominator lessthan those ofBm+lyfrom(8), (9),andthedefinition ofBnastheright- hand sideof(7),weseethat Similarly En>Bnforallvalues ofw>m. Tji 1From(10)weprove Lt-fr^r*-Thispieceofwork isso n-><tin** similar tothecorrespondingwork attheend ofArt.108thatwe leave itasanexercise forthestudent. CO HenceVUJRj11isconvergentifR<R. Therefore within the circle |x \=*Rland within theregion specifiedforc, Thisshows that2anxc+nsatisfies Weierstrass's M-test foruniform convergence (Bromwich,Art.44),asRl95,andtheE'&areallinde- pendentofc. Thiscompletestheproofthat2^>n=2o nc+flsatisfies allthe conditionsspecified,sothedifferentiation withrespecttocisnow justified.This holds within thecircle\x\^R^WecantakeR^ great enoughtoinclude anypointwithin thecircle\x\=R. Iftheroots oftheindicialequationareequalinstead ofdiffering byaninteger,theonlydifference intheabove work isthatais nottobereplaced by&(c-/3),asno(c-/3) cannowoccur inthe denominator ofan. [ForasupplementtoChaps. IX.andX.seeArts. 171-177. They dealwithregular integrals,Fuchs' theorem, ordinary andsingular points, equationsofFuchsiantype,characteristic index, normal and subnormalintegrals.] CHAPTER XI ORDINARY DIFFERENTIAL EQUATIONS WITH THREE VARIABLES, ANDTHECORRESPONDING CURVES AND SURFACES 111.Weshallnow consider somesimpledifferentialequations expressing propertiesofcurves inspaceand ofsurfaces onwhich these curveslie,orwhichtheycutorthogonally (asinElectro- statics theEquipotentialSurfaces cuttheLines ofForce ortho- gonally). Theordinary*differentialequationsofthischapterare closelyconnected with thepartialdifferentialequationsofthe next. Beforeproceedingfurther thestudent should revise hissolid geometry. Weneed inparticularthefactthatthedirection-cosines ofthetangenttoacurve are (dx dydz \dsW? ds' i.e.areintheratiodx:dy:dz. Simultaneous linearequations with constant coefficients have already been discussed inChapterIII. 112.Thesimultaneous equations ^=^r=^-TheseequationsrQR expressthat thetangenttoacertain curve atanypoint (x,y,z) hasdirection-cosinesproportionalto(P,Q,R).IfP,Q,andRare constants, wethusgetastraight line, orrather adoublyinfinite systemofstraight lines, asonesuch linegoesthrough anypointof space. If,however, P,Q,andRarefunctions ofx,y,and z,weget asimilarsystemofcurves, anyoneofwhichmaybeconsidered as generated byamoving point whichcontinuouslyalters itsdirection *i.e.notinvolving partial differentia] coefficients. 133 134 DIFFERENTIAL EQUATIONS ofmotion. TheLines ofForce ofElectrostatics form such a system.* Bz.(i). -- .................................. (1) Obviousintegralsare x-z=a,.................................... (2) y--6 ..................................... (3) theequationsoftwoplanes, intersectingintheline which bysuitable choice ofthearbitrary constants aandbcanbemade togothrough anygiven point, e.g.through (/,g,h)ifa=f-h and 6-0-A.. Instead ofpickingoutthesinglelineofthesystem thatgoesthrough onegiven point, wemaytake theinfinityofsuch lines that intersect agiven curve, e.g.thecircle x2+2/2=4,2=0. Theequationsofthis circle, takentogether with(2)and(3),give andhence a*+62=4..................................... (5) This istherelation thatholds between aandbiftheline istointer- sectthecircle. Eliminatingaandbfrom(2), (3),and(5),weget theelliptic cylinder formed bythose lines ofthesystem which meet thecircle. Similarly thelines ofthesystem which meet thecurve 0(s,y)-o, s=o form thesurface<f>(x-z,y-z)=0. Ex.(ii).7-7?-^..................................(6) Obviousintegrals are a?2-fz2 a,....................................(7) V=b........................................ (8) arightcircularcylinder andaplane that cuts itinacircle. The differentialequations thereforerepresentasystem ofcircles, whose centres alllieontheaxis ofyandwhoseplanesare allperpen- dicular tothis axis. Onesuch circle goesthrough anypointofspace.Thatthrough (/>9,h)isX2+22^2+h* 9y=sgt Asurface isformed bythe circles ofthesystemthat intersect a given curve. *The equationsofthe lines offorce aredx/^- =dy/'S-^d*/S^ twhereyisthepotential function. Io* /W /d* ORDINARY EQUATIONS WITHTHREE VARIABLES 135 Ifthegiven curve isthehyperbola --'.-. (7)and(8)give,foracircleintersectingthishyperbola, xa=a, y=b, a6a andhence Eliminatingaand6from(7), (8),and(9),wegetthehyperboloid ofonesheet, X2+zz 2 formed bythose circles ofthesystem that intersect thehyperbola. Similarly, starting from thecurve 0(z2 ,t/)=0, 2=0,wegetthe surface ofrevolution 0(x2 -f-22 ,t/)=0. 113. Solution ofsuch equations bymultipliers. If dx_dy_dz ~p-~Q-~R' each ofthese fractions isequalto Idx+mdy+ndz ~~ Thismethod maybeusedwithadvantageinsome examplesto obtain azerodenominator andanumerator that isanexact differential, oranon-zero denominator ofwhich thenumerator isthe differential. Ex (i)dx=dy=dZ 'v;z(x-y) x2+y*' ^.f. xdx-ydy-zdz xdx-ydy-zdzEach fraction = .-- --r~\ nr~' *"\=-A--- xz(x+y)-yz(x-y)- z(x*+yz ) therefore xdx-ydy-zdz=0, i.e. #2y2-z2=a. Similarly ydx+xdy-zdz=*Q, i.e.2xy-zz=*b. .I+yl+x z dzdx+dydx-dy = --==-- , z2+x+y y-x giving log2=log(2+o;+2/)+loga= -log(x-!/) +log6, t.0. a(2+x+y)=&/(-y). 136 DIFFERENTIAL EQUATIONS Examples forsolution. Obtain thesystemofcurves, denned bytwoequationswith an arbitrary constant ineach, satisfying thefollowing simultaneous dif- ferentialequations. Interpret geometrically wheneverpossible. ^-^ ^? (9\^dydz xyz* mz-ny nx-lz ly-mx' /rt. dx dy dz ..dxdvdz '*),. .2~2=ikr.= r*^'(V~ z~w^* dx dydz xdx dydz (5)===(6)~z^ o=ssa. y+zz+xx+yz2-2yz-y2y+zy-z (7)Find theradius ofthecircle ofEx.2thatgoesthroughthe point (0,-n,m). (8)Find thesurface generated bythecurves ofEx.4that intersect thecircle y2+z2=1,xQ. (9)Find thesurfacegenerated bythelines ofEx. 1that intersect thehelix x2+y2=r2 ,z=*ktan-1- x (10)Find thecurve whichpasses throughthepoint (1,2,-1)and issuch that atanypointthedirection-cosines ofitstangentareinthe ratio ofthesquaresoftheco-ordinates ofthatpoint. 114.Asecond integral found bythehelpofthe first. Consider the equations dxdydz Anobviousintegralisy+2x=a (2) Usingthisrelation, weget dx_dz T~3z2sma' givingz-x3sina=6. Substitutingfora,z-x*ain(y +2#)=6 (3) Is(3)really anintegralof(1)? Differentiating (3), {dz-3x2dxsin(y+2x)}-*cos(y+2x).{dy+2dx} 0, which istrue invirtue of(1).So(3)isanintegral. Examples forsolution. dxdydz/0dxdy dz 1"* 3"Sz+tanfy-Sx)' %**-%**> ^ xyy2 ORDINARY EQUATIONS WITHTHREE VARIABLES 137 115.General and special integrals ofsimultaneous equations. li w=aandv~b aretwoindependent integralsofthesimultaneous equations fafyfa T~Q=R' then<j>(u,v)=0 representsasurfacepassing throughthecurves of thesystem, andshould thereforegiveanother solution, whatever theform ofthefunction</>. Ananalytical proofofthis isreserved forthenextchapter,as itsimportance belongs chieflytopartialdifferentialequations. <f>(u,v)=0iscalled theGeneralIntegral. Some simultaneous equations possess integralscalledSpecial, which arenotincluded in theGeneralIntegral. Examples forsolution. (1)IntheEx. ofArt.113u~x*-y*-z* andv~2xy-z2 ,sothe GeneralIntegralis0(z2-t/2-2 ,2xt/-3a)0. Thestudent should verifythisinthesimplecases where <f>(u,v)~u-v or<t>(u,v)= (2)Verifythat fortheequation dxdydz " theGeneralIntegral maybetaken as while z**x+yisaSpecial Integral. 116.Geometrical interpretation oftheequation Pdx+Qdy+Rdz =0. This differentialequation expressesthatthetangenttoacurve isperpendiculartoacertainline,thedirection-cosines ofthistangent andlinebeing proportional to(dx,dy,dz)and(P,Q,R)respectively. Butwesawthatthesimultaneousequations dx_dy _dzP^-R expressedthatthetangenttoacurve wasparalleltotheline(P,Q,R).Wethusgettwo sets ofcurves. Iftwocurves, oneofeachset, intersect, theymust intersect atright angles. Nowtwocases arise. Itmayhappenthattheequation Pdx+Qdy+Rdz~Q isintegrable. Thismeans thatafamilyofsurfaces canbefound, allcurves onwhich areperpendiculartothecurvesrepresented by 138 DIFFERENTIAL EQUATIONS thesimultaneousequationsatallpointswhere these curves cutthe surface. Infact, this isthecasewhere aninfinite number ofsurfaces canbedrawn tocutorthogonallyadoublyinfinite setofcurves, asequipotentialsurfaces cutlines offorce inelectrostatics. Onthe other hand, thecurvesrepresented bythesimultaneousequations maynotadmit ofsuch afamilyoforthogonalsurfaces. Inthis casethesingle equationisnon-integrable. Ex.(i).Theequation integratesto afamilyofparallel planes.WesawinEx.(i)ofArt.112thatthesimultaneousequations dx^dy_dzT'T^T representedthefamilyofparallellines x-ay-bz___ -. Theplanes aretheorthogonal trajectoriesofthelines. Ex.(ii). zdx-xdz=09 ,dxdz . i.e. -0x z integratesto z=ex, afamilyofplanes passing through theaxis ofy.WesawinEx.(ii)ofArt.112thatthecorrespondingsimultaneous equations dx_dy __dz ~z=S !)*:S~x representedasystemofcircles whose axes allliealong theaxis ofy, sotheplanesaretheorthogonal trajectoriesofthecircles. Examples forsolution. Integrate thefollowing equations, andwheneverpossible interpret theresultsgeometrically andverify thatthesurfaces aretheorthogonal trajectoriesofthecurvesrepresented bythecorresponding simultaneous equations: (1)xdx+ydy+zdz**Q. (2)(y2+z2-x*)dx- 2xydy-2xzdz-0.[Divide byx2 .] (3)yzdx+zxdy+xydz=*Q. (4)(y+z)dx+(z+x)dy+(x+y)dz**Q. (5)z(ydx-xdy)**y*dz. (6)xdx+zdy+(y+2z)dz=0. 117.Method ofintegration when thesolution isnotobvious. When anintegrable equationoftheform ORDINARY EQUATIONS WITHTHREE VARIABLES 139 cannot besolved byinspection, weseek forasolution byconsidering firstthesimplercasewhere zisconstant andsodz=0. Forexample, yzdx+2zxdy -3xydz=Q becomes,ifzisconstant, giving xy2=a. Asthiswasobtained bysupposingthevariable ztobeconstant, itisprobablethat thesolution oftheoriginal equation canbe obtained byreplacingtheconstant abysome function ofz,giving xy*=f(z) leadingto y2dx+2xydy-~-dz=0. This isidentical with theoriginal equationif Jf y22xy __dz yz2zx -3xy df3xy* 3/(z)f.e.-y-=*-=---- 9dz z z df_3dz givingthefinal solution xy2=cz3 . Foraproofthat thismethod holds goodfor allintegrable equations,seeArt. 119. Examples forsolution. (1)yzlogzdx-zxlogzdy+xydz=0. (2)2yzdx+zxdy-xy(l+z)dz=Q. (3)(2x*+2xy+%xz*+l)dx+dy+2zdz**Q. [N.B. Assume xcon- stant atfirst.] (4)(y2+yz)dx+(zx+z2 )dy+(y2-xy)dz=0. (5)(x2y-y3-y2 z)dx+(xy2-x2z-a?3 )dy+(xy2+x2 y)dz=0. (6)Show that theintegralofthefollowing equation representsa familyofplanes with acommon line ofintersection, andthat these planesaretheorthogonal trajectoriesofthecircles ofEx.2oftheset followingArt.113 : (mz-ny)dx+(nx- Iz)dy+(ly-mx)dz=0. 118. Condition necessary foranequation tobeintegrable. If Pdx+Qdy+Rdz=Q...........................(1) hasanintegral ^(xyy,z)=c,which ondifferentiationgives -dg+?fdz-0,ydz 140 DIFFERENTIAL EQUATIONS a^ a-L a^ then Multiply equations (2), (3),and(4)byP,Q,andRrespectively, andadd.Weget Iftheequation (1)isintegrable,thiscondition mustbesatisfied. Thestudent familiar with vectoranalysiswillseethat ifP,Q,R arethecomponentsofavector A,thecondition maybewritten A.curlA=0. Ex.Intheworked exampleofthelast article, yzdx+2zxdy-3xydz=0, Theconditiongives yz(2x+3x)+2zx(-3t/- 1/)-3xy(z-2z) =0, i.e.5xyz-Sxyz-fSxyz 0, which istrue. Examples forsolution. (1)Show that theequationsinthe lasttwo sets ofexamples satisfythiscondition. (2)Show that there isnosetofsurfacesorthogonaltothecurves givenby efo dydz zx+yI' *119.Thecondition ofintegrability issufficient aswellasnecessary. We shall provethat thecondition issufficient byshowingthat when itissatisfied themethod ofArt.117willalways besuccessful ingivingasolution. Werequireasalemma thefactthat ifP,Q,Rsatisfythecon- dition, soalsodoP1=XP,Ql=XQ,R1=Xfi, where Xisanyfunction ofx,y,and z.Weleave thisasanexercise tothestudent. Tobeomitted onafirstreading. ORDINARY EQUATIONS WITHTHREE VARIABLES 143 Themethod ofprocedureistoeliminate oneofthevariables and itsdifferential, sayzand<fe,from these twoequations andthe differ- ential ofthesecond ofthem. Differentiating (2), 2dx-dy-dz=*Q. Multiplying byxandadding to(1), (y+2x)dx+(z-x-y)dy=*Q, orusing (2), (y+2x)dx+(x-2y~l)dy=0, whichgives xy+x*-y2-yc2............................... (3) Thus thecurves ofthefamilythat lieintheplane (2)arethesections bythatplaneoftheinfinite setofrectangular hyperbolic cylinders (3). The result ofthisexample could have beenexpressed bysaying thattheprojections ontheplaneofxyofcurves which lieintheplane (2)andsatisfy equation (1)areafamilyofconcentric, similar and similarlysituatedrectangular hyperbolas. Examples forsolution. (1)Show that there isnosingle integralofdz=*2ydx-fxdy. Prove thatcurves ofthisequationthat lieintheplanez=xylie alsoonsurfaces ofthefamily (x-l)2(2y-1)=c. (2)Show thatthecurves of that lieontheellipsoid liealsoonthefamilyofconcentric spheres x2+y2+z2=k*. (3)Find theorthogonal projection ontheplaneofxzofcurves which lieontheparaboloid3z=x2+y2andsatisfytheequation 2dz=(x+z)dx+ydy. (4)Find theequationofthecylinder,with generators parallelto theaxis ofy,passing through thepoint (2,1,-1),andalsothrougha curve that liesonthespherexz-fy2+z2**4and satisfies theequation (xy-f2xz)dx-fy2dy+(#a-fyz)dz=0. MISCELLANEOUS EXAMPLES ONCHAPTER XI. dx<fydzm {'~ (4)xy' y*x-2x* 2y*-x*y$z(x*~y*Y dz 144 DIFFERENTIAL EQUATIONS (5)(2x+y* +2xz)^+2xy^+x*ji*~l. (6)Findf(y)iff(y)dx-zxdy-xy logydz isIntegrable. Find thecorresponding integral. (7)Show thatthefollowing equationisnotintegrable: 3ydx+(z-3y)dy+xdz*=Q. Prove that theprojection ontheplaneofxyofthecurves that batisfy theequation and lieintheplane 2x+y-~z=aaretherectangular hyperbolas X2+3^_^2_^^^ (8)Find thedifferentialequationsofthefamilyoftwisted cubic curves y=ax2 ;y2=bzx.Show that allthese curves cutorthogonally thefamilyofellipsoids (9)Find theequationsofthecurve thatpasses throughthepoint (3,2,1)andcutsorthogonally thefamilyofsurfaces x+yz=c. (10)Solve thefollowing homogeneous equations byputting xuz, y=*vz: (i)(xz- 1/2-21-f2xy4-2xz)dx4-(y*-z2-x2+2yz+2yx)dy (ii)(2xz-yz)dx4-(2yz-xz)dy-(za- (iii)z2dx+(z2-2yz)dy+(2^/2-yz-xz}dz=0. (11)Prove that iftheequation Pldxl isintegrable, then 9P.dPt\(dP, dP,\dPr dP. t~ dx.)'\dxr~ dxt where r,5,tareanythree ofthefour suffixes 1,2,3,4. Denotingthisrelation byCr8t=Q,verify that ^234~^2<?i34+PzCiz*~^123=identically, showingthatonly three ofthese four relations areindependent. Verify thatthese conditions aresatisfied fortheequation (Xj3-Z23z4)dx1+(x23-x tza:r4)dx.2 4-(x33-x^x^ dji'z+(#43-^0^2X3) dx=0. (12) Integrate theequationofEx.11bythefollowing process: (i)Supposea?3andx4constant, andthusobtain a?!44-24~^XyX 2xzx^=a. (ii)Replaceaby/(a? 8,o?4).Bydifferentiation andcomparison with dfdf theoriginal equation obtain~ ,~,andhenceyandthesolution a?!44-a?244-a?844-a;44-4g1x>8&4-o. MISCELLANEOUS EXAMPLES U6 (13)Integrate theequationofEx.11byputting x^ux^x2^vxit XsWXi> (14)Show that thefollowing equationsatisfies theconditions of integrability andobtain itsintegral: ysinwdx+xsinwdy-xysinwdz-xycoswdw=0. (15)Show thattheequation adx*+bdy*+cdz*+2fdydz+2gdzdx+%hdxdy^Q reduces totwoequationsoftheform Pdx+Qdy +Rdz^Q if ale+2fgh-of2-lg2-ch2=>0.(Cf.aresult inConies.) Hence show thatthesolution of xyz(dx2+dy2+dz2 )+x(y2+z2 )dydz+y(z2+x2 )dzdx +t(x2+y2)dxdy~Q is(x2+y*+z*-c) (xyz- c)=0.(Cf.Art.52.) (16)Show thatthecondition ofintegrabilityof Pdx+Qdy+Rdz=Q...........................(1) implies theorthogonalityofany pairofintersecting curves ofthe families dx/P^dy/Q^dz/R..............................(2) A 1and dx--5- ^---5"=\^~-............. dzdy/yl\dx SzJ l\dy dxj' Hence show thatthecurves of(3)alllieonthesurfaces of(1). Verifythisconclusion forP~ny-mz, Q=lz-~nx, R*=mx-ly. (Forthesolutions ofthecorresponding equations,seeearlierexamples inthischapter.) (17)Thepreceding example suggeststhat ifa=const., /3=const. aretwointegralsofequations (3),theintegralofequation (l)should beexpressibleintheform/(a, /3)= const., andhence that Pdx+Qdy +Rdz should beexpressibleasAda+Bd/3,whereAandBarefunctions of aandft. Verify that forthecase Q=-zxlogz, R=xy, A**-ft, and Z?=a. Hence obtain anintegralof(1)intheforma=Cj&, i.e. f/=cxlogz. [ForasupplementtothechapterseeArts. 168-170. Theydealwith anintegratingfactor forhomogeneous equations,andwithMayer's CHAPTER XII PARTIAL DIFFERENTIAL EQUATIONS OFTHEFIRST ORDER. PARTICULAR METHODS 121.Wehavealready (inChap. IV.)discussed theformation oi partialdifferentialequations byelimination ofarbitraryfunctions orofarbitraryconstants. Wealsoshowed how incertainequations, ofgreat importanceinmathematicalphysics, simple particular solutions could befound bytheaidofwhich morecomplexsolutions could bebuiltuptosatisfysuch initial andboundary conditions as usually occur inphysical problems. Inthepresent chapter weshall beconcernedchieflywithequa- tions ofgeometrical interest, andseek forintegralsofvarious forms, u general,"" complete," and" singular," and theirgeometrical interpretations. Exceptional equationswillbefound topossess integralsofanother form called" special." 122.Geometrical theorems required. Thestudent should revise thefollowingtheorems inanytreatise onsolidgeometry: (i)Thedirection-cosines ofthenormal toasurface/(x, y,z)=0 atthepoint (x,y,z)areintheratio #.#.#.dx' dy' dz* Since dfjdfdz ,df/dfdz-* <*-=*-=P>sav>and -*- /*-=-*-= <7>say,3xidzdx^ J dyldzdy* J' thisratiocanalsobewritten p:q:-1. Thesymbols pandqaretobeunderstood ashere defined all throughthischapter. (ii)Theenvelopeofthesystemofsurfaces f(x, y,z,a,6)-0, 146 PARTICULAR METHODS 147 where aand6arevariableparameters,isfound byeliminating aandbfrom thegiven equationand The resultmay contain other locibesides theenvelope (cf. Chap. VI.). 123.Lagrange's linear equation and itsgeometrical interpretation. This isthenameappliedtotheequation Pp+Qq-R,..............................(1) where P,Q,Rarefunctions ofx,y,z. Thegeometrical interpretationisthat thenormal toacertain surface isperpendiculartoalinewhose direction-cosines areinthe ratioP :Q:R.Butinthelastchapter wesawthatthesimultaneous equations dxdydz/0, T-^-72.................................(2) representedafamilyofcurves such thatthetangentatanypoint had direction-cosines intheratioP:Q:R, andthat </>(u yv)=ff (where u=const, andv=const, were twoparticular integralsof thesimultaneousequations) representedasurface throughsuch curves. Through every pointofsuch asurfacepassesacurve ofthe family, lying whollyonthesurface. Hence thenormal tothe surface must beperpendiculartothetangenttothiscurve, i.e. perpendiculartoalinewhose direction-cosines areintheratio P :Q:R.This isjustwhat isrequired bythepartialdifferential equation. Thusequations (1)and(2)areequivalent,fortheydefine the same setofsurfaces. When equation (1)isgiven, equations (2)are called thesubsidiary equations. Thus (f>(u,v)=0 isanintegralof(1),ifu=const, andv=const. areanytwoindependentsolutions ofthesubsidiary equations (2) and <f>isanyarbitraryfunction. This iscalled theGeneralIntegral ofLagrange'sLinearEquation. Ex.(i). p+q~l. Thesubsidiary equationsarethose discussed inEx.(i)ofArt. 112, viz. dxdydz T=T~P representingafamilyofparallel straightlines. 148 DIFFERENTIAL EQUATIONS Twoindependent integralsare x-z-a, y-z=*b, representing twofamilies ofplanes containingthesestraightlines. Thegeneral integralis<f>(x-z, t/-2)=0, representingthesurface formed bylines ofthefamily passing throughthecurve 0(3,y)=0,2=0. Ifwearegivenadefinite curve, such asthecircle o?2 -f*/2=4,2=0, wecanconstruct acorresponding particular integral theelliptic cylinder formed bylines ofthefamily meetingthegiven circle. Ex.(ii).zp~-x.[Cf.Ex.(ii)ofArt. 112.] Thesubsidiary equations are dx_^dy dz " ofwhich twointegrals arex2+z2a,yb. Thegeneral integral (x*+z2 ,t/)=0 represents thesurface of revolution formed bycurves(circles inthiscase)ofthefamilyinter- secting thecurve(^y)=o,2=0. Ex.(in).Find thesurfaces whosetangent planes cutoffanintercept ofconstantlength kfrom theaxisof2. Thetangent plane at(x,yyz)is Z-z=p(X-x)+q(Y-y). PuttingZ=F=0,Z=z-px-qy=k. Thesubsidiary equations are dxdy dz xyz-k9 ofwhich y=ax,z-k=*bx, areintegrals. Thegeneral integral 0R,-J=0 represents anycone with its\x x' vertex at(0,0,k),andthese surfacesclearly possessthedesiredproperty. Examples forsolution. Obtaingeneral integralsofthefollowing equations. [Cf.the first setofexamplesinChap.XL] (1)xp+yq=z. (2)(mz-ny)p +(nx-lz)q^ly-mx. (3)(y*+z2-x*)p-2xyq-f2^2=0. (4)yzp+zxq=*xy. (5)(y+z)p +(z+x)q**x+y. PARTICULAR METHODS 149 (6)(z*-2yz~ (7) 2?+32=5-3+tan(?/-3;e). (8)zp-zq=z2+(y+x)*. (9)Find asolution ofEx.(1)representingasurface meetingthe parabola */2=4,r,z=l. (10)Find themost generalsolution ofEx.(4)representingaconicoid. (11)Show that ifthesolution ofEx.(6)representsasphere,the centre isattheorigin. (12)Find thesurfaces allofwhose normals intersect theaxis ofz. 124. Analytical verification ofthegeneral integral. Weshallnow eliminate thearbitraryfunction </>from(w,t;)=0, and thus verify analyticallythat this satisfies Pp-\-Qq^R, providedu=aand v=6aretwoindependent* integralsofthesubsidiary equations dx_dydz p"g"fl- Differentiate<^>(w,v)=0 partiallywithrespecttox,keeping y constant;zwillvaryinconsequenceofthevariation ofx.Hence wegetfy(dududz\ d/dv Svdz\ du\dx dzdx/ dv\dxdzdx)' deb/du du\ d<f>/dv dv\- i.e.-*r(cr+P^-)+*(* +P--T =0.du\dx*dz/ dv\dx*dz/ ri. .,, dfh/du du\ dc/>/dv dv\- Snarly-(^+9%)^(Sy+9&=' Eliminatingtheratio-^:-^from these lasttwoequations, du du\fdv dv\ /du du\/dv dv /dudv_dudv\ /dudv__dudv\ \dydzdzdy/*\dzdxdxdz/^ dudvdudv ^'dydx -o^f du *du3du *~Butfromu=a,---dx+~-ay+---dz=0,dx dyydz andhence from thesubsidiary equations,ofwhich u=aisanintegral, a-a--a--dx^ dydz Ifuandvarenotindependent, (^^-^^ )andtheothertwosimilar expressionsailvanish identically (Edwards' Differential Calculus, Art. 610),which reduces equation (1)to0=0 160 DIFFERENTIAL EQUATIONS -11 T>^ Ti ~ Similarly P^-+#=-+#~--0.Jdx dydz Hence p/i 7?' \ "" \9vdzdzdy)' \dzdxdxdz/'\&eByBydx/' so(1)becomes Pp+Qq*=R 9theequation required. 125. Special integrals. Itissometimes stated that allintegrals ofLagrange'slinearequationareincluded inthegeneral integral (w,v)=0.But this isnotso. Forinstance, theequation hasassubsidiary equations dx_dy_dz Thuswemaytakeu=x+y,v=x- ^/z,andthegeneral integralas <f>(x+y, z-Vz)=' Butz=0 satisfies thepartialdifferentialequation, thoughitis obviously impossibletoexpressitasafunction ofuand v. Suchanintegraliscalledspecial.Itwillbenoticed that inall theexamples givenbelow thespecial integralsoccur inequations involvingaterm which cannot beexpandedinseries ofpositive integral powers. Inarecentpaper M.J.M.Hill* hasshown that ineverycase wherespecial integralsexisttheycanbeobtained byapplyinga suitable method ofintegrationtotheLagrangian systemofsub- sidiary equations (seeExamples5and6below). Healsounder- takes there-classification oftheintegrals,thenecessityofwhich taskhadbeenpointedoutbyForsyth.f Examples forsolution. Show that thefollowing equations possessthegiven general and special integrals: (1) (2) (3) [Chrystal.] *Proc.London Math. Soc. 1917. tProc.London Math. Soc. 1905-*. PARTICULAR METHODS 15 (4)Byputting (z-x-yy^winCrystal's equation (Ex. 3),obtaii [dwdw ~\2(1-fiv)^--f2=-+1HO.ox oyJ Thisshows that zxy=*Qisasolution oftheoriginal equation. [Hill. (5)Show that theLagrangian subsidiary equationsofChrystal' equation (Ex. 3)maybewritten dx4fdz anddeduce that T-(2-E-2/)= -(z xy), ofwhich 2-x-?/=Oisaparticularsolution.[Hill. (6)Obtain thegeneral andspecial integralsoftheequation byimitatingHill's methods asgiveninExs. 4and 5. 126.The linear equation with nindependent variables. Th( general integraloftheequation wherePi=^,pz^-,...etc.,and theP'sandRarefunction* OX^ uX^ oftheX'Bandz,is^(w^w2,w3,...wn)=0, where%=const., w2=const.,...etc.,areanynindependent integrals ofthesubsidiary equations dx_dx2_dxs_ __dz 'P\-P 2-P~---R- Thismaybeverified asinArt. 124. Thestudent should write outtheproofforthecase ofthree independentvariables. Besides thisgeneral integral, special integralsexist forexcep fcionalequations, justasinthecase oftwoindependentvariables. Examples forsolution. (2) (3)(Xi-x (4) (5) (6) 152 DIFFERENTIAL EQUATIONS *w *sp%* 127.Theequation P--_--fQ*--fRn-0. IfP,Q,72arefunctionsox oy <?z ofx,y,zbutnoof/,theequationcanbeviewed fromtwodifferent aspects. Consider, forexample, =0............................ (1) Wemay regardthis asequivalenttothethree-dimensional equation p-q=Z/z,.................................(2) ofwhich0(x+y, x-i/z)=Qisthegeneral integralandz=0 a special integral. Ontheother hand, regarding (1)asanequationinfour variables, wegetthegeneral integral which isequivalentto/=^(x +y,x-\/z),where\jsisanarbitrary function, but if Thus/=zisnoanintegralof(1),although/== certainly givesasolution. Ingeneralitmaybeprovedthat regardedasfour-dimensional, where P,Q,Rdonotcontain/,has nospecial integrals.* Asimilar theorem istrue foranynumber of independentvariables. Examples forsolution. (1)Verify that if/=z,/=0isasurfacesatisfying andhence that this differentialequation, interpretedthrec-dimension- ally,admits thethreespecial integralsx=0,y=0,2=andthegeneral integral <f>(^/z-\/x,*Jz- <\/y)=0. (2)Show that thegeneral integralofthelastexample represents surfaces throughcurves which, iftheydonotgothroughtheorigin, cither touch theco-ordinateplanesorliewhollyinoneofthem. [Hint. Prove thatjr^Jt-- ),andthatdjo/ds=Qifz-0, unless x,y,zareallzero.] *SeeAppendix B. PARTICULAR METHODS 168 (3)Show that\Jx~-4-\/y~-=0,regarded two-dimensionally, repre- sents afamilyofparabolas \/y \/x+c,and theirenvelope,the co-ordinate axes o?=0, t/=0;while regarded three-dimensionallyit represents thesurfaces z=(f>(y- a;'). 128. Non-linear equations. Weshallnow considerequationsin whichpandqoccur other than inthe firstdegree.Beforegiving thegeneral method weshall discuss foursimplestandard forms, for which a" complete integral" (i.e.oneinvolving twoarbitrary constants) canbeobtained byinspectionorbyothersimple means. InArts. 133-135 weshallshowhow todeducegeneral andsingular integralsfrom thecomplete integrals. 129.Standard I.Only pandqpresent. Consider, forexample, thisequation q=3p2 . Themostobvious solution istotakepandqasconstantssatisfying theequation, sayp=a,q=3a2 . Then, sincedz=--pdx +qdy=adx+ 3a2dy, z=ax4-3a2y4-c. This isthecomplete integral, containing twoarbitrary constants aand c. Ingeneral,thecomplete integralof/(p, q)=0is zax~4-by4-c, where aand6areconnected bytherelation/(a, b)=*0. Examples forsolution. Findcomplete integralsofthefollowing: (1) 2>=224-l. (2) ;>a4-ja=l. (3)p=&.(4)pV=l. (5)^2_ 52==4<(6)pq^p +q. 130.Standard II.Only p,q,andzpresent. Consider theequation Asatrial solution assume that zisafunction ofx-\-ay (-w,say),where aisanarbitraryconstant. ml dzdzdudz dzdzdu dzThen T?=--=-_-. -~=^- ;q=^=, -=a-^-. c/#aw (toawctyaaa// aw */dz\2 Substitutingin(1),z2 (-j-)(2*4-a2 )=1, i.e.9(4-ay+&)*-=(^24-a*)8 . 164 DIFFERENTIAL EQUATIONS Ingeneral,thismethod reduces/(*, y,#)=0totheordinary differentialequation dz dz Examples forsolution. Findcomplete integralsofthefollowing: (1)4z=*pq. (2)z2=I*?2*?*. (3)q*=zy (1-p). (4)p*+q*=27*. (5)p(z+p)+q-0. (6)p*=zq. 131.Standard HI. f(x,p)=F(y, q).Consido*: theequation p-3.T2=(f-y. Asatrial solution puteach side ofthisequation equaltoan arbitraryconstant a,giving p=3x*+a ; <?=\/(y+a). But dz=pdx+qdy =(3x2-fa)rfx+^/(y+a) dy ; therefore =x3+ax+(y4-a)*+6, which isthecomplete integral required, Examples forsolution. Findcomplete integrals ofthefollowing: (1)p*= (3)yp= (5)pev=qex .(6)q(p-cosa?)=cosy. 132. Standard IV. Partial differential equations analogous toClair- aut's form. InChap.VI.weshowed thatthecomplete primitiveof wasy=cx +/(c),afamilyofstraightlines. Similarlythecomplete integralofthepartialdifferential equation isz=ax+by +/(a, 6),afamilyofplanes. Forexample,thecomplete integralof z=px+yy-fpa+ga is z=ax+by-faa+62 . Correspondingtothesingularsolution ofClairaut's form, giving theenvelopeofthefamilyofstraight lines,weshall findinthenext PARTICULAR METHODS 155 article a" singular integral"ofthepartialdifferentialequation^ givingtheenvelopeofthefamilyofplanes. Examples forsolution. (1)Prove that thecomplete integralofz*=px+qy-%p- 3qrepre- sents allpossible planes throughthepoint (2,3,0). (2)Prove that thecomplete integralofz=px+qy+<\/(p2+q*+1) representsallplanesatunitdistance from theorigin. (3)Prove that thecomplete integralofz=px+qy+pq/(pq-p-q) representsallplanes such thatthealgebraic sum oftheinterceptson thethree co-ordinate axes isunity. 133. Singular Integrals. InChap.VI.weshowed that ifthe familyofcurvesrepresented bythecomplete primitiveofanordinary differentialequationofthe firstorderhadanenvelope,theequation ofthisenvelope wasasingularsolution ofthedifferentialequation. Asimilar theorem istrueconcerningthefamilyofsurfacesrepre- sented bythecomplete integralofapartialdifferentialequationof the first order. Iftheyhaveanenvelope,itsequationiscalled a " singular integral." Toseethat this isreallyanintegral wehave merelytonotice that atanypointoftheenvelopethere isasurface ofthefamily touchingit.Therefore thenormals totheenvelope and thissurface coincide, sothevalues ofpandqatanypointof theenvelopearethesame asthat ofsome surface ofthefamily,and thereforesatisfythesameequation. Wegavetwomethods offinding singular solutions, namely from thec-discriminant andfrom thep-discriminant, andweshowed that these methods gavealso node-loci, cusp-loci, and tac-loci, whose equationsdidnotsatisfythedifferentialequations. Thegeometrical reasoningofChap.VI.canbeextended tosurfaces, butthe dis- cussion oftheextraneous lociwhich donotfurnishsingular integrals ismorecomplicated.*Asfarastheenvelopeisconcerned, the student whohasunderstoodChap.VI. willhave nodifficultyin understandingthat thissurface isincluded amongthose found by eliminatingaand6from thecomplete integral andthetwoderived equations /(x, y,z,a,6)=0, i i *Seeapaper byM.J.M.Hill, Phil. Trans. (A),1892. 156 DIFFERENTIAL EQUATIONS orbyeliminating pandqfrom thedifferentialequationandthe twoderivedequations F(z,y, *,P>q)=0, dF-0 dpu' !r-- 3q Inanyactualexample oneshould testwhether what isapparently asingular integral reallysatisfies thedifferential equation. Ex.(i).Thecomplete integraloftheequationofArt.132waa Differentiating withrespecttoa,=x 4-2a. Similarly 0=> y+2b. Eliminatingaandb, 4z=-(x2-f?/2 ). Itiseasilyverified that this satisfies thedifferential equation andrepresentsaparaboloidofrevolution, theenvelopeofalltheplanes represented bythecomplete integral. Ex.(ii).Thecomplete integraloftheequationofArt.130was 9(z+a</+6)2=(z2+a2 )3.......................... (1) Differentiating withrespecttoa, 18y(x+ay+b)=6a(z*+a2 )*......................(2) Similarly 18(x+ay+b)=Q..................................... (3) Hence from(2), a-0 ..................................... (4) Substituting from(3)and(4)in(1),z=0. But2=0givesp==0,andthese values donotsatisfy thediffer- entialequation 22(p2z2-fg2 )=1. Hence2=0 isnotasingular integral. Ex.(iii).Consider theequation p* zq. Differentiating withrespecttop,2^=0. Similarly=2. Eliminating pandqfrom these threeequations, weget 2=0. This satisfies the differential equation,soitreallyisasingular integral. .But itisderivable byputting 6=0 in which iseasily found tobeacomplete integral. Sos0 isbothasingular integral andaparticularcase ofth6 complete integral. PARTICULAR METHODS 167 Examples forsolution. Find thesingular integralsofthefollowing: (1)z=px+qy4-logpq. (2)z~px-fqy+pz-fpq+ga . (3)z=px+qy+\p2q\ (4)z=>px+qy+p/q. (5)z=pq. (6)za=l+7?2+?2 -(7) 2>3+?3=27*. (8)Show thatnoequation belongingtoStandard I.orIII.hasa Ingular integral. [The usualprocessleads totheequation 1.] (9)Show thatz**Q isboth asingular integral andaparticularcase facomplete integralofq2=z2pz(lp*). 134.General Integrals. Wehave seen, inEx.(i)ofthe last rticle, that alltheplanes represented bythecomplete integral z=az-t-6iy +a2+62...........................(1) ouch theparaboloidofrevolution represented bythesingular ntegral 42=3-(z2+2/2 )...............................(2) Now consider, notalltheseplanes, butmerely thoseperpendicular otheplane y=0. These arefound byputtingb=in(1),giving >fwhich theenvelopeistheparabolic cylinder 4z=-a*..................................(3) Take another set,those whichpassthroughthepoint (0,0,1). From(1), l~a2+62 , (1)becomes z=axy\/(l -a2 )+1, >fwhich theenvelopeiseasily found tobetheright circular cone (z-l)2-*^2............................(4) Ingeneral, wemayput6=/(), where/isanyfunction ofo, [iving z=az+?//(a)+a2+{/(a)}2......................(5) Theenvelopeof(5)isfound byeliminating abetween itand heequationfound bydifferentiatingitpartially withrespecttoa, i.e.0=x+yf'(a)+2a+2f(a)f(a)................ (6) If/isleftasaperfectly arbitrary function, theeliminant is ailed the" general integral"oftheoriginaldifferentialequation. Equations (3)and(4)areparticular integralsderived from the ;eneral integral. Wemaydefine thegeneral integralofapartialdifferential quationofthefirstorder astheequation representingtheaggregate 1theenvelopesofevery possible singly-infinitesetofsurfaces that P.D.H. If 158 DIFFERENTIAL EQUATIONS canbechosen outofthedoubly -infinitesetrepresented bythe complete integral.These setsaredefined byputting6/(a)is thecomplete integral. Itisusually impossibletoactually performtheelimination of abetween thetwoequations givingtheenvelope,onaccount ofthe arbitraryfunction /anditsdifferential coefficient. Thegeometrical interest lieschieflyinparticularcases formed bytaking/assome definite (and preferably simple)function ofa. 135. Characteristics. Thecurve ofintersection oftwo con- secutive surfacesbelongingtoany singly-infinitesetchosen from thoserepresented bythecomplete integraliscalled acharacteristic. Now such acurve isfound from theequationofthefamilyof surfaces bythesametwoequationsthatgivetheenvelope.For instance, equations (5)and(6)ofthelast article, foranydefinite numerical values ofa,/(a),and/'(a),define astraightline(asthe intersection oftwoplanes),and thisstraightline isacharacteristic. Thecharacteristics inthisexampleconsist ofthetriply-infiniteset ofstraightlines thattouch theparaboloidofrevolution(2). Theparabolic cylinder (3)isgenerated byonesingly-infiniteset ofcharacteristics, namelythoseperpendiculartotheplane t/=0, while thecone(4)isgenerated byanother set,namelythose that passthroughthefixedpoint (0,0,1).Thusweseethatthegeneral integral representstheaggregate ofallsuchsurfaces generated bythe characteristics. Ifasingular integral exists, itmust betouched byallthechar- acteristics, andtherefore bythesurfaces generated byparticular setsofthemrepresented bythegeneral integral.Itiseasilyverified thattheparabolic cylinderandrightcircular cono ofthelastarticle touch theparaboloidofrevolution. 136. Peculiarities ofthelinear equation. Todiscuss thelinear equation Pp+Qg=R(1) onthese lines, supposethat uconst. and v=const, aretwoindependent integralsofthesubsidiary equations.* Then itiseasilyverified thatanintegralof(1)is u+av 4-6=(2) 11Since uandvareindependent, atleast oneofthemmust contain z.Let thisonebeu.Wemake thisstipulationtoprevent u+av+bbeing afunction of xandyalone, inwhich caseu+av+bQwould make terms in(1)indeterminate, instead ofdefinitely latisfyingitintheordinary wa* PARTICULAR METHODS 159 Thismay betaken asthecomplete integral. Thegeneral integralisfound from w+flw+/=0,..............................(3) *+/'()=<>...............................(4) From(4),aisafunction ofvalone, say a=F(v). Substitutingin(3), u=afunction ofv, say ti-^Mv), which isequivalenttothegeneral integral </(w,v)=0 found atthe beginningofthechapter. The linearequationisexceptionalinthat itscomplete integral (2)isaparticularcase ofthegeneral integral. Anotherpeculiarity isthatthecharacteristics, which areherethecurvesrepresented by thesubsidiary equations,areonly doubly-infiniteinnumber instead oftriply-infinite. Onlyonepasses throughagiven point (ingeneral), whereas inthenon-linear case, exemplified inthelast article, an infinite number maydoso,formingasurface. Examples forsolution. (1)Find thesurface generated bycharacteristics of that areparalleltotheaxis ofx.Verify that itreallysatisfies the differentialequation andtouches thesurfacerepresented bythesingular integral. (2)Prove that 3a=4a?yisanintegralof representing theenvelopeofplanes included inthecomplete integral andpassing throughtheorigin. (3)Prove that thecharacteristics of</=3/)2thatpassthrough the point (-1, 0,0)generatethecone(x-f-l)2-f12?/z=-0. (4)What isthenature oftheintegral (?/-f1)2+4a?z=Ooftheequation z-px+qy+pfal (5)Show that either oftheequations ax+by, maybetaken asthecomplete integralofacertain differentialequation, andthattheothermaybededuced from itasaparticular case ofthe general integral. [London.] 160 DIFFERENTIAL EQUATIONS (6)Show that z(x+a)2ebyisacomplete integralofthedifferential equation p*=4zeqyf*. (xy\2~v~- )ispartofthegeneral integralofthe"~ y* sameequation, anddeduce itfrom theabovegiven complete integral, [London.] MISCELLANEOUS EXAMPLES ONCHAPTER XII. (1)z~px+qy-p2 q. (2)=^x-fqy-(px +z)2 q. (3)z(z2+xy)(px-qy)~x*. (4)p*-q*=3x-2y. (5)P12^2x2p2+x^p^0. (6) (7)p*+q*-3pqz~Q. (8) (9)Pi+Pz+Ps^te. (10) (11)z2p2y+6zpxy+2zqx*+4x2y==Q. (12) (13)p2z2+q2~p2 q. (14)(z-px- (15)Find theparticular case ofthegeneral integralof thatrepresentstheenvelopeofplanes included inthecomplete integral andpassing through thepoint (1,1,1). (16)Prove that iftheequation Pdx-fQdy-f-Rdz=isintegrable,it representsafamilyofsurfaces orthogonal tothefamily represented by Hence findthefamily orthogonalto (17)Find thesurfaces whosetangent planesallpassthrough the origin. (18)Find thesurfaces whose normals allintersect thecircle -z2+2/2=4, *=0. (19)Find thesurfaces whose tangent planes form with theco- ordinate planesatetrahedron ofconstant volume. (20)Prove that there isnonon-developable surface such that every tangent planecuts offintercepts from theaxeswhosealgebraicsum iszero. (21)Show that iftwosurfaces arepolar reciprocals withrespect to thequadricx2+y2=*Zz,and(x,y,z),(X,Y,Z)aretwocorresponding points (oneoneach surface) such thatthetangent plane ateitherpoint isthepolar planeoftheother, then Hence show that ifonesurface satisfies /(z, y>2>^>?)=0, theother satisfies /(P.Q.PX+QY- Z,X,Y) 0. (These equations aresaid tobederived from each other bythe Principle ofDuality.) MISCELLANEOUS EXAMPLES 181 (22)Show thattheequationdual to b giving z~PX+QY-Z=-XY. Hence derive (asanintegralofthefirstequation) z=-xy. (23)87means ofapartialdifferential equationeliminate the arbitraryfunction from theequation [Differentiating partiallywithrespecttoxandyyweget and l+q Hence (1+<p)(y+zq)=(1+q)(x+zp), or( (24)Usethemethod ofEx.23toverifythesolutions oftheexamples onp.148. (25)Findparticular integralsofthefollowing partialdifferential equations torepresentsurfacespassing through thegivencurves. (i)p+q=l;x=0, y2=z.(ii) (iii)(y-z)p +(z-x)q=x-y;2=0, y (iv)x(y- (v)yp~ (vi)(y- [Eliminate x,y,zfrom thetwoequations ofthecurve andtwo independent integrals u(x, y,z)=a, v(x, y,z)=bofthesubsidiary equations. Thisgivesarelation between aand b.Replace abyu(x,y,z), bbyv(x, y,z),andwegettheintegral required. Thus for(i)u(x, y,z)zsx-z=a, v(x, y,z)^y-z=^b. (Cf. p.148.) From these andthecurve equations x=0, y*=z,wegeta=*-y2 , fc^y-y2 ,BO(6-a)2=-a. Replaceabyx-z,bbyy-2,andwegettheintegral (y-x)2=z-x. Similarlyfor(ii), (iii),and(iv). In(v)and(vi)weeliminate x>y,z,I from fiveequations. Answers,(ii)yz~(x +y)*. (iii) (iv) (x+y+z)*=27xyz. (v) (vi)8- *CHAPTER XIII PARTIAL DIFFERENTIAL EQUATIONS OFTHEFIRST ORDER. GENERAL METHODS 137.We shallnowexplain Charpit's method ofdealingwith equations withtwoindependentvariables andJacobi's method for equationswithanynumber ofindependentvariables. Jacobi's method leadsnaturallytothediscussion ofsimultaneouspartial differentialequations. Themethods ofthischapterareconsiderably morecomplicated than those ofthe last.Weshall thereforepresentthem intheir simplest form, andpass lightlyover severalpointswhich might be considerably elaborated. 138. Charpit's fmethod. InArt.131wesolved theequation j>-3s=g-y (1) byusinganadditional differentialequation p-3z2=a, (2) solvingforpandqinterms ofxandy,andsubstitutingin dz^pdx+qdy, (3) which thenbecomesintegrable, considered asanordinarydifferential equationinthethree variablesx,y,z. Weshallnowapplyasomewhat similar method tothegeneral partialdifferentialequationofthe firstorder withtwoindependent variables F(x,y,z,p,q)=0 (4) Wemust findanotherequation, say f(x, y,z,p,y)=0, (5) *Tobeomitted onafirstreading. fThis method waspartly duetoLagrange, butwasperfected byCharpit. Charpit's memoir waspresented totheParisAcademyofSciences in1784, but theauthor diedsoonafterwards andthememoir wasneverprinted. 162 GENERAL METHODS 163 such thatpandqcanbefound from(4)and(5)asfunctions of x,y,zwhich make(3)integrable. Thenecessary and sufficient condition that(3)should bein- tegrableisthat nfdQdR\~fdR dP\^fdP dQ\ A,., x. .P(-a- )+Q(a--a-+#(-a-~a^H (identically),\9z 3y/ \9x 9z/ \3y 9z/v 7" where P^P)Q=qy#=-1, Bydifferentiating (4)partiallywithrespecttox,keeping yand zconstant, butregarding pandqasdenoting thefunctions ofx, y,25obtained bysolving (4)and(5),weget 9qdx Similarly +.o .........................(8) Jox From(7)and(8),J=-,...........................(9) v' v"dxBxdpdpdxv' , T. ,,whereJstands for ^-~--^-- . ag>op _. .. .T SimUarly J- '.....................<10) dz (/Zu(JoQ(/Z Substitutingin(6)multiplied by*/,weget (3T7I~\f3XT'3f\ /3TP^f3f"^f\OJ!Of OJuOf\ /OJjOf OJ*Of\ dzdpdpdzJ^\dzdq dqdz/ dydq dqdy dxdp dpdx' _ydFdf (dF dF\df* J A Ifn ^.Q JA- dpdx dqdyvdp* dq/dz *Jcannot vanish identically,forthiswould implythatFand/,regarded as functions ofpandq,were notindependent. This iscontrary toourhypothesis thatequations (4)and (5)canbesolved forpandg. 164 DIFFERENTIAL EQUATIONS This isalinear equationoftheform considered inArt. 126, with x,y,z,p,qasindependentvariables and/asthedependent variable. Thecorresponding subsidiary equationsare &Jy-dz___&_____ dq_ df(U) ~~d]?~ 3F~ dFdF~dFd]F~~dFdF'v ' 'dp ~dq~Pdpq dq~dx+Pdzdy+?dz Ifanyintegraloftheseequationscanbefound involving por qorboth, theintegral maybetaken astheadditional differential equation (5),which inconjunctionwith(4)willgivevalues ofp andqtomake(3)integrable.This willgiveacomplete integralof (4),from whichgeneralandsingular integralscanbededuced in theusual way. 139.Asanexampleoftheuseofthismethod, consider the equation 2xz-px*-2qxy +pq=Q.........................(1) Takingtheleft-hand side ofthisequationasFyandsubstituting inthesimultaneous equations (14)ofthelast article, weget dx__dy_dz_dp_dj_<tf x2~-q~ 2xy^p ~~px*~+2xyq -2pq~~ 2z~-%~ 1)~ 0~' ofwhi(5h anintegralisq=*a.....................................(2) From(1)and(2), p777 2x(z-ay)dx ,Hence dz^pdx+qdy=^^-fady,X Q> dz-ady __2xdx*/" *~"n >z-ay xz-a i.e.z^ay +b(x*-a). This isthecomplete integral.Itiseasytodeduce theSingular Integral z=z2 y. Theform ofthecomplete integralshows that(1)could have beenreduced to z=PX -fqy-Pq, which isaparticularcase ofastandard form, bythetransformation ov T>dz Idz jc2=Z;P=*^=;r-^~.dX%xdx Equationsthatcanbesolved byCharpit's method areoften solved moreeasily bysome such transformation. GENERAL METHODS 165 Examples forsolution. Apply Charpit's method tofindcomplete integralsofthefollowing: (1)2z+;p2-f^+2?/2=0.(2)yzpz=q. (3)pxy+pq+qy**yz. (4)2x(z2q2+l)=*pz. (5)q=$p*. (Cf.Art.129.) (6)z*(p*z* +q*)=l.(Cf.Art. 130.) (7^7>-3z2=92-?/. (Cf.Art.131.) (8)z=*px+qy+p* f?a .(Cf.Art. 132.) (9)Solve Ex.2byputting */2=F,z2=Z. (10)Solve Ex.4byasuitable transformation ofthevariables. 140.Three ormore independent variables. Jacobi's* method. Consider theequation F(x ltx2,z3,pltp2,p3)=0,........................(1) where thedependentvariable zdoesnotoccurexcept byitspartial differential coefficients pl9p2,p3withrespecttothethreeindependent variables xl,x2,x3.Thefundamental idea ofJacobi's method is verysimilar tothat ofCharpit's. Wetrytofindtwoadditionalequations Fifa, **>*a>Pi,V*PB)= <*i>........................(2) jF2(z!, Xa,a?3,ft,ft,p3)=a 2........................(3) (where axanda2arearbitrary constants), such thatp^p2,pzcan befound from(1), (2),(3)asfunctions ofxltx2,x$thatmake dz=p 1dxi+p 2dx2+psdx 3........................(4) integrable,forwhich theconditions are dxl Now,bydifferentiating (1)partiallywithrespecttoxlfkeeping x2andx3constant, butregarding pltp2,p3asdenotingthefunctions ofxltXK$3obtained bysolving (1), (2), (3),weget dFdFdPldFdp2dFdp3X-T^-7i--T^-X"~ ~T^-^-U.............. .....lUl dxldpldxdp2dxldp3dxlv' .., ,dFldFldpldF^p 23Fldp3A... Similarly a-1** 5+aiF+i" /^................... (?) JOXldplOXLOp2OX1OptOX) *CarlGustav Jacob Jacobi ofPotsdam (1804-1851) may beconsidered aserne ofthecreators oftheTheoryofElliptic Functions. The"Jacobian"or"Func- tional Determinant"reminds usofthelarge partheplayedinbringingdeter- oiinante intogeneraluae. 166 DIFFERENTIAL EQUATIONS From(6)and(7), 9(^A) ,W *i)??2 , , 9l),. ..MT,.y,BFdF ldFwhere~ -^denotes the Jacobian =----;$-"3(a^,ft) a^j3ft9^ Similarly and|+> i+l,(10) Addequations (8), (9),(10). Twoterms are d(F, F-L)dpzd(F, FI)dpl Similarly twootherpairsofterms vanish, leaving \+^ . ^ _x+ _ +t_l=0 a^iayxa^?iaa^ a^2a^2a^2a^2a#3ap3a^3a^3* Thisequationisgenerallywritten as(F,FJ=0. Similarly (F,F t)=0 and(^,^=0. Butthese arelinearequationsoftheform ofArt. 126. Hence wehave thefollowingrule : Trytofindtwoindependent integrals,.P1=a1andF^=a 2Jofthe subsidiary equations do^ dpldx2dp% dx<i Ifthesesatisfythecondition andifthep'scanbefoundasfunctions ofthex'sfrom F-F^Oi-Ft-a^O, integratetheequation* formed bysubstitutingthesefunctionsin dz *Foraproof that thisequation willalways beintegrable, seeAppendix C, GENERAL METHODS 167 141.Examples onJacobi's method. Ex.(i). tyjX&t +Sp&S+pSp^O......................... (1) Thesubsidiary equationsare dxl dpldx2 dp2dx3 dp3 -2xjX 32;^-3x3*-2p2p3""-p2*2^ ofwhichintegralsare -^i^i^i ^^i*................................. (2) and F2?==p 2*=a2.................................. (3) Nowwith these values (Fl9F2)isobviously zero, so(2)and(3)can betaken asthetwoadditionalequations required. Hence dz=a1x1~ldx1+a2dx2-a2~2(2a 1x3-f3a2x32 )dx3 or 2;=axlog a?! thecomplete integral. Ex.(ii). ( Thisequationisnotoftheform considered inArt. 140,asitinvolves z.Butput dz dx. du Idu _._-- where u isanintegralof(4). Similarly p2--P2/P4;p,--P3/P4. (4)becomes (z2+^)(^2 +^3)2-^A^4=0,..................... (5) anequationinfourindependent variables, notinvolvingthedependent variable u. Thesubsidiary equationsare axl_dPl^dxz___dP2 _____2__ ofwhichintegralsare Fl^Pl=aly....................................(6) *V^P 2-P3=2,..............................(7) F*=x*Pi-<**.................................. (8) Wehave tomake surethat(Fr,F8)=0,where rand sareanytwo oftheindices 1,2,3.This iseasilyseen tobetrue. Solving (5), (6), (7), (8),weget P!-^; P4=asa?r1 ;2P2-a2V{<W(* 2+*3)};P3=P-a; BO du=ajdx1-fa3x^~ldx^+\a2(dx2-dx3) W{aia3/(x2+z3)}(dx2-fdx3), i.eua^j+aslogx44-Jaaf^a~x*)^V{aias(2;2+^s)}+a * 168 DIFFERENTIAL EQUATIONS Sow=*0gives, replacing z4byz,a1/a3byAvJa2/a3byA2,aja 3by48, logz+A1x1+A2(x2-zs)\/{^i(2 +^3)}+^8^0, thecomplete integralof(4). Examples forsolution. ApplyJacobi's method tofindcomplete integralsofthefollowing: (1)Pi8+PB+Pa-l. (2) (3)p&t+ptx^pa*. (4) (7)p (8) ( 142.Simultaneous partial differential equations. Thefollowing examplesillustrate sometypicalcases : Ex.(i). F^p^ +p^x^^Q,........................... fl) ^l=Pl +P2^2=0..................................... (2) Here f1-^ Thus theproblem maybeconsidered asthesolution oftheequation (1),with partofthework (thefindingofFJalready done. ThenextstepistofindF%such that (F,Fj=0=(Fl9Fj. Thesubsidiary equationsderived byJacobi'sprocessfromFare dx^ dpi dx% dp2 dx$ dp$ Pl""- Anintegralis p1==a.....................................(3) WemaytakeF2aspltsince this satisfies (F,F2) (FltFJ. Solving (1), (2), (3)andsubstitutingindz=pldxl+p2dx2+p9dxB, dza(Zxj,-ax2~~1dx2-fax9~2dx3) BO z=a(x1-logx2x3~l )+b. Ex.(ii). 1^1*1 +^2-^-0,...........................(4) *is JJl-pt+p,-l-0............................(5) Here (F,J1)=p l-fpa(-l)=Pi-p a- Thismust vanish iftheexpressionfordzistobeintegrable. Hence wehave theadditional equation 7>i-P 8=0.................................. (6) Solving (4), (5), (6)andsubstituting, 2 f-log GENERAL METHODS 169 Inexamplesofthistypewedonothave tousethesubsidiary equations. The result hasonly onearbitrary constant, whereas in Ex.(i)wegottwo. Ex.(iii). JFaaV +flCs'+Pa-O,...........................(7) ^i~?i+P2 +z32=3 ............................(8) Here (F,FJ)=2x 1+2x2-2x3. Asxl9x2,xsareindependent variables, thiscannot bealwayszero. Hence wecannot findanintegrable expressionfordzfrom these equations, which havenocommonintegral. Ex.(iv).jFfsp 1+p1+j>a1-31-3a?a-4x82=0,.................. (9) Fl^xlp1-x2p2-2xl*+2x2*=0,........................ (10) JF.ssp,- 238-0.............................................. (11) Solving (9),(10), (11)andsubstitutingintheexpressionfordz, dz=(2x l-fx2)dxl+(xl+2x2)dx2+2x3dx3, so z=X)2-fXfl2+x2*4-x32+a. Thistime there isnoneed towork out(F,Fj 9(F,Fj,(FltF2). Ex.(v). F^pt+pz-l-x^O,...........................(12) Sj-Xj-O,...........................(13) Thesegivedz=x2dxl+dx2-fx^dx^. Asthiscannot beintegrated,thesimultaneousequations have no commonintegral. Ex.(vi). Fzsxipi-x&i +pa-pi^Q,........................(15) JFjM^+pj-^-Xg-O............................... (16) Here (F,FJ-^-xx(-1)-p2+x2(-1)-^-p2+xl-x2. AsinEx.(ii),thisgivesusanewequation F2^p l-p2+xl-x2^0.......................... (17) Now (F,I^-P!-*!-^-!)*^-!)- ^i=0, and (F19^2)=(~l)-l-l-(-l)r-l)-(-l)=0, BOwecannot getanymoreequations bythismethod. Thesubsidiary equationsderived fromFare dxl_dp l_dx2_dp2_dx3_dpB_dx^_dp^ -aj1"~^ 1""x2~-p 2~-l"""" 1"~0* Asuitable integralisF3=p3=a, .................................(18) forthis satisfies (F,Fj-(F l9FJ-(F F3)=0. Wehavenowfourequations (15), (16), (17), (18). Thesegive BO 170 DIFFERENTIAL EQUATIONS But inthisexample wecanobtain amore general integral. The twogiven equations (15)and (16)and thederived one (17) are equivalenttothesimplerset : Pi-x*.................................... U9) Pi-*i,.................................... (20) P3-?4=0...................................... (21) From(19)and(20),zXLXZ4-anyfunction ofx3andx4. (21)isalinearequationofLagrange's type,ofwhich thegeneral integralis tf>(z,z3+z4)=0, i.e.zisanyfunction of(#3+#4),andmayofcourse alsoinvolve xl andx2. Hence ageneral integralofallthreeequations,orofthetwogiven equations,isz=x^+^(^+x^ involving anarbitraryfunction. Thecomplete integral obtained by theothermethod isincluded asaparticularcase. Thegeneral integral could have beenobtained from thecomplete,asinArt. 134. Examples forsolution. Obtain common complete integrals (ifpossible)ofthefollowing simultaneousequations: (1)Pi'+V-Sfri +a^'-O. (Pi-Pz)(xi~xz)+^3*3-1=0. (2)xSptp^xJptfi-Xtfpjpi**!. (3)PiP2P3-^xix^3ss^W22:3^3 -2^4=0, p2+y)3-2x2-2z3=0.2j>j-p2=0. (5)Pix3*+p3=0, (6) (7) (8)Find thegeneral integralofEx.(5). (9)Find thegeneral integralofEx.(7). MISCELLANEOUS EXAMPLES ONCHAPTER XIII. (1)2*^3^^3 +2^2=0.(2) (3)Qx&tfi (pa+pa)- 4/>42-0, (4)^zp^ +p3)-4-0, (5) (6)p^-^-y^-^-y^ (7)Findasingular integralofz representingtheenvelopeofallthehyper-surfaces (inthiscasehyper- planes)included inthecomplete integral. (8)Show thatnoequationoftheformF(x ltx2,a?3,pl9p%tp8)t- hasasingular integral. MISCELLANEOUS EXAMPLES 171 (9)Show that ifzisabsent from theequation F(x, ytz,p,9)=0, Charpit's method coincides with Jacobi's. (10)Show that ifasystemofpartialdifferential equationsislinear andhomogeneousinthep'sandhasacommon integral Z==a1w1+a2w2+..., where thew'sarefunctions ofthe x's,thenamoregeneral integralin 2=^>(M 1,W2,...). Find ageneral integralofthesimultaneousequations (11)ItPiand;?2arefunctions oftheindependentvariables xl9 satisfyingthesimultaneousequations F(xvx2,pl9^2)=0=^(05!,sca,pl9pz), provethat (F 9Fl}+ -f&)-0.^ Hence show that ifthesimultaneousequations, taken aspartial differentialequations,have acommonintegral, (F,F-^^0isanecessary butnotasufficient condition. Examine thefollowing pairsofsimultaneousequations: (i)Fsj 7)(77* 7P^ [Here ^-7-^-=identically, andtheequations cannot besolved 90>i>P) forplandp2.] r)(FJ?\ [Here (F, jPj)and^--^-bothcome tofunctions which vanish when thep'sarereplaced bytheir values interms ofxland#2There isnocommon integral.] (iii)F==Pl-p2*+x2=Ot [These have acommon integral, although ^4^comes toa d(Pi> P*l function thatvanishes when the^'sarereplaced bytheirvalues.] NoteonCharpit's Method (pp.162-164). Sometimes wecanfindanequation /(#, y,z,p,q)= which isan integral, notofthesubsidiary equations (14),butofsimpler equations obtained from thesebyusing theoriginaldifferentialequation (4.).This willsatisfy (13),notidentically, butinvirtue of(4),and inconjunction with(4)will stillmake(3)integrable. Thus inEx. 2,Art. 139,pz=a'iB anintegral, notofdzl(-2yzp2+q)=dplyp*, butofdz[(-yzjt)= giving finallytheresuJA onp.xvi.SimilarlyforJacobi's method. CHAPTER XIV PARTIAL DIFFERENTIAL EQUATIONS OFTHESECOND ANDHIGHER ORDERS 143.We shall firstgivesomesimple examplesthatcanbe integrated byinspection.After thisweshall dealwith linear partialdifferentialequationswith constant coefficients;these are treated bymethods similar tothose used forordinarylinearequations withconstant coefficients. Therestofthechapterwillbedevoted tothemore difficultsubjectofMonge's*methods. Itishopedthat thetreatment willbefullenoughtoenable thestudent tosolve examples andtomakehimbelieve inthecorrectness ofthemethod, butadiscussion ofthetheorywillnotbeattempted.f Several exampleswilldealwiththedetermination ofthearbitrary functions involved inthesolutions bygeometricalconditions. J Themiscellaneousexamplesattheendofthechaptercontain severalimportantdifferentialequations occurringinthetheoryof vibrations ofstrings, bars,membranes, etc. Thesecond partial differential coefficients ^~t~~-,^-s will iA *,3u , * idxdxdlJdVbedenotedbyr,s,trespectively. 144.Equations thatcanbeintegrated byinspection. Ex.(i). s=2x+2y. Integrating withrespecttox(keeping yconstant), \Similarly, integrating withrespecttoy, <f>(y)dy +f(x), sayz-x*y+xy*+f(x)+F(y). *Gaspard Monge,ofBeaune (1746-1818), Professor atParis, created Descriptive Geometry. Heapplieddifferential equations toquestionsinsolid geometry. tThestudent who desires thisshould consult Goursat, Surrintegration de* equations^ auxderiveespartielles dusecond ordre. tFrost's Solid Geometry, Chap.XXV., maybereadwithadvantage. 172 SECOND ANDHIGHER ORDERS 173 Ex.(ii)Find asurface passing throughtheparabolas 3=0, t/2=4o&and 3=1,y2**-4.ax9 andsatisfyingxr+2p**Q. The differentialequationis giving Ju Thefunctions /andFaretobedetermined from thegeometrical conditions. Putting20andx= 7/2/4a, Similarly Hence , 1 t/and^^o"^""'28aa; i.e.Saxz=4aa?- 1/2 ,aconicoid. Examples forsolution. (1)r=6x.(2)jrys-1. (3)*=sma:i/. (4)o>r-fp=Qx2 !/3 . (5) t/5+p=cos(x+?/)~t/sin(x +7/). (6)t-xq=xz . (7)Findasurfacesatisfyings=8xyandpassing throughthecircle (8)Find themost general conicoid satisfying (9)Find asurface ofrevolution thattouches 2=0and satisfies (10)Find asurfacesatisfying J^Gz3 ?/,containing thetwo lines t/=0=2,y=l=2. 145.Homogeneous linear equations with constant coefficients. In Chap.III.wedealt atsomelengthwith theequation (Di+ajD*-1+aJD~* +...+<Oy -/(),...............(1) where D&ZJ-.dx P.D.E. R 174 DIFFERENTIAL EQUATIONS Weshallnow dealbrieflywith thecorresponding equationin twoindependent variables, (Dn+a^-W +aa7)-2J9'a+...+anD'*)z=f(x, y),......(2) whereD^| ,D^~ . oxoy Thesimplestcase is(D-mD')z =0, i.e.p-mq0 9 ofwhich thesolution is<f>(z,y+mx) =0, i.e.z**F(y+mx). Thissuggests,what iseasily verified, that thesolution of(2) if/(a,y)=0is z^F^y-fm^)+F2(y+m 2x)+...+J?n(?/+wwz), where thew^m2,...mnaretheroots(supposedalldifferent)of mn4-om*1-1-fo^m"-24- ...4-an=0. Theroots ofm3-3m2+2?n=-0 are0,1,2. Hence z=F^y)-f-Fa(y+x) Examples forsolution. (1)(D3-6DaD'+llDZ)'t-6D')2i-0. (2)2r+55+2<=0.(3)l^-f-^-x7ax2 a?/2 (4)Find asurface satisfyingr-f.9=andtouching theelliptic paraboloidz=4x2+?/2alongitssection bytheplane y2x+1.[2V.B. Thevalues ofp(andalso ofq)forthetwosurfaces must beequalfor anypoint ony**2x+l.] 146. Casewhen theauxiliary equation hasequal roots. Consider theequation (D-mD')2z=Q ...............................(1) Put (D-mD')z=u. (1)becomes (D-mD')u-0, givingu=F(y+mx) ; therefore (D-wZ)')2=F(y+ mx), or p-m^=F(y+mx). Thesubsidiary equationsare dx_dy dz SECOND ANDHIGHER ORDERS 175 gvng and dz-F(a)dx~Q, i.e. z-xF(y+mx)=6, BOthegeneral integralis <j){z-xF(y +mx),y+wx}=0orz=xF(y +mx)+F1(y+mx). Similarly wecanprovethattheintegralof is z=*xn~lF(y+mx) +xn~2Fl(y+mx)+...+Fn^(y+mx). Examples forsolution. (1)(4D2+12DD' +9D'a)z=0. (2)25r- 40s+l&=0. (3)(D*-D*D'+lDD'*)z==>0. (4)Find asurfacepassing through thetwo lines 2J=x= ) 2i-l=z~2/=0, satisfying r-4s -f4=0. 147.TheParticular Integral. Wenowreturn toequation (2)of Art. 145,andwrite itforbrevityas F(D,D')z=f(x,y). Wecanprove, following Chap.III.stepbystep,thatthemost generalvalue ofzisthesum ofaparticular integral andthe complementaryfunction (whichisthevalue ofzwhen thediffer- entialequation has/(x, y)replaced byzero). Theparticular integral maybewrittenjrrn~~ y\-f(x, y),and J.\*J)I') wemaytreat thesymbolicfunction ofDandD'aswedidthat of Dalone, factorising it,resolvingintopartial fractions, orexpanding inaninfinite series. IQx* sothesolution of(Z)2-6DD' +9Z)/2 )2-12x2+3Qxy s Examples forsolution. (1) (2) 176 DIFFERENTIAL EQUATIONS (3)Find arealfunction Vofxandy,reducingtozerowheny=0 andsatisfying S2y^y 148.Short methods. When/(x, y)isafunction ofax+by, shorter methods maybeused. NowD<t>(ax+by)= a<j>'(ax+by) ;D'^>(ax+&*/)= b<f>'(ax+6y). Hence jP(D, 1)') (ax+by)=F(a,6)(H)(ax-f6y), where(n)isthenthderived function of0,nbeingthedegreeof F(D, D'). Conversely -^"**^'.........(A) provided F(a,&)=/=0, e.^._1_ -sin(2x+3y)_ '*COS(JiC+^}"2^*: 2 .3T4.273* since^(2x+3y)maybetaken as-sin(2x-f3y)if <t>'" (22?+3y)-cos(2x+3y). Todealwiththecasewhen F(a, b)=0,weconsider theequation (D-mD')z=p-mq=xr \}s(y-fmx), ofwhich thesolution iseasily found tobe xr+l z=^i BOwemaytake Hence (B) e'9'D*-WD'+D'tan(y while 111(4a:+^=^'T'8in ~cos -ja;cos(4x+y)by(B). SECOND ANDHIGHER ORDERS 177 Examples forsolution. (1) (2) (3) (4)2r--3* =5e*/e*. (5) +-L (6)4r- 149.General method. Tofindageneral method ofgettinga particular integral,consider (D-mD')z=p-mq=f(x, y). Thesubsidiary equationsare -ty-dz ~T~~~^m~~f(x 9yY ofwhich oneintegralisy+mx=c. Usingthisintegraltofindanother, dz=f(x, c-mx)dx t z" If(x>c~~mx)&x"*"constant, where cistobereplaced byy+mxafterintegration. Hence wemaytakej^- ^>.f(x ty)as \f(x,c-mx) dx,where cisreplaced byy+mxafter integration. Ex. (D-2D')(D +D')z=(y-l)e*. Here \f(x,c-2x)dx=\ (c-2x-l)exdx=(c-2x+l)e*. Therefore^-.,.(y-l)ex **(y+l)ex 9replacingcbyy+2x. Similarly-=? .(y-hi)e*isfound from 1 byreplacingcbyy-x, giving ye*,which istheparticular integral required. Hence z*yex+<j>(y+2x)+\ls(y- x). Examples forsolution. (1)(Z)2+2DZ)/-fZ)/2)22cost/~a;smy. (2)(D*-2DD'-l5D'*)z=*l2xij. (3) (5)r-<=tan3xtan y-tanxtan3 y. im?^4?^.^-iwa*a x' 178 DIFFERENTIAL EQUATIONS 150.Non-homogeneous linear equations. Thesimplestcase is i.e.p-mq^az y giving </>(ze~ax ,y+mx)=0, or z=eax \[s(y+mx). Similarly wecanshow thattheintegralof (D-mDf-a)(D-nD'--b)z=0 is z=eaxf(y+mx)+ebxF(y+nx), while that of(I)~mU-a)2z=0 is z=eaxf(y+mx)-fxeaxF(y+mx). Buttheequationswhere thesymbolical operator cannot be resolved intofactors linear inDandDrcannot beintegratedinthis manner. Consider forexample (D2-D')z=0. Asatrial solution patz^=ehx+kv ,giving Soz=#(x+hv}isaparticular integral, andamoregeneral one is hv\where theAandhineachterm areperfectly arbitrary, andanynumber oftermsmaybetaken. Thisform ofintegralisbestsuited tophysical problems,aswaa explainedatsomelengthinChap.IV. Ofcourse theintegralof anylinearpartialdifferentialequationwith constant coefficients maybeexpressedinthismanner, buttheshorter formsinvolving arbitraryfunctions aregenerallytobepreferred. Examples forsolution. (1)DD'(D-2D'-3)z=Q. (2) (5)(2J94~3Z)2D/+D/2)2=0. (6)~~ oxoy (7)(J9~2Z)/ (8)Find asolution ofEx.(4)reducingto1whenx=+00and to t/2when z=0. 151. Particular Integrals. Themethods ofobtaining particular integralsofnon-homogeneous equationsareverysimilar tothose in Chap. III., soweshallmerely giveafewexamples. Ex.(i). 23-3.2. SECOND ANDHIGHER ORDERS 179 Hence *-- where A8-Zhk+li+1-0. Ex.(ii). (DfD'- {1H-D-fD'+terras ofhigher degree} (D+2D'^ ,,.., \xn+--fterms ofhigher degreer .L4D+5D'A... .,\^^j1_l hterms ofhigher degree j- Acting on4-f3x+6y,thisoperator gives Hence 2-6+x+2/y+e*f(y-x)-f<**F(y-2x). Ex.(Hi). (D*-DD'-2D)z=*am(3x "Sn 3-f2D .sm3-27)" TVsin(3cc+4?/)+T2Tcos(3-c-f-4?/), Hence z=yVsin(3x+y)+T2Tcos(3x+4y)- where A2-M-2/i-0. Examples forsolution. (1)(D~D'-l)(7)~Z)'-2)2=e2a;-y. (2)s+p-5=2 +i/. (3)(D-Z (6)(Z)-3Z)/ 152.Examples inelimination. Weshallnow consider theresult ofeliminating anarbitraryfunction from apartialdifferential equationofthe first order. Ex.(i). 2px-qy=<j>(x*y). Differentiating partially,firstwith respecttoxandthen tot/,weget 2rx-sy+2p=> 2xy<j>'(x2 y)t and*2sx-ty-q=x2 <}>'(x2 y), whence x(2rx-sy+2p)=2y(2sx-ty-q) or 2#2r-bxys-\2y2t+2(px+qy) 0, which isofthe firstdegreeinr, ,t. 180 DIFFERENTIAL EQUATIONS Thesameequationresults fromeliminating \fsfrom Ex.(ii). j>2+g Thisgives 2pr+s=20'(2a; +t/), and 2ps+0'(2z -fy), whence 2pr-fs=4ps+2, againofthe firstdegreeinr, ,f. Ex.(Hi). y-p=</>(x-q). Thisgives-r**(1-s)0'(z-y), and l-s"-^0'(ic~?)> whence rJ=(l-s)a or 2s+(r*-s2 )=l. Thisexamplediffers from theother two inthatpandqoccur in thearbitraryfunction aswell aselsewhere. The result contains a term in(rt-$) Examples forsolution. Eliminate thearbitrary function from thefollowing: (1)W-?+3y2=0(2z+y2 )- (2)--0(*). (3)p+x-y=<f>(q-2x +y). (4) (5)p*-x=<f>(q*-2y). (6) 153. Generalisation ofthepreceding results. Ifuand t;are known functions ofx,y,z,p,q,andwetreat theequationu= <f>(v) asbefore, weget du dudu du(dv dvdv dv rA-+5^--f-+p^~=rw-+*--+;r-+A- 3p 3ydx^dz \9p dqdx^dz , 9?/ 9u9w du(dv dvdv dv\ ,..and *a'+^a~+a-+?^-=(* a-H-^a-+aT+?a-) -0 (v)-9p 9?dy*dz \dp dqdy*dz/^^' Eliminating 0'(v)wefindthattheterms inrsand scancel out, leavingaresult oftheform\ ,,. ).c&(v),JYv; where R,S,T,UandVinvolvep,y,andthepartialdifferential coefficients ofuand t;withrespecttox,y,z,p,q. m, . TrdudvdvduThe coefficient U=a-5-~-=-, opctydp9g wAicA vanishesifvisafunction ofx,y,zonlyandnotofporq. These results willshow uswhat toexpect whenwestart with theequationsofthesecond orderandtrytoobtainequationsofthe ftrsfcorder fromthem. SECOND ANDHIGHER ORDERS 181 154.Monge's method ofintegrating Rr+Ss+Tt=V. We shall owconsiderequationsofthefirstdegreeinr,s,t,whose coefficients !,S9T,Varefunctions ofp,q,x,t/,z,andtrytoreverse theprocess EArts. 152and153. Since dp=--dx+^dy=*rdx ad dq=sdx+tdy, ecomes R +Ss+T i.e.Rdpdy +Tdqdx-Vdydx-s (Rdy2-Sdydx +Tdx2)=Q. The chief feature ofMonge's method isobtainingoneortwo Nations betweenp,q,x,y,z(each relationinvolvinganarbitrary inction)tosatisfythesimultaneousequations Rdpdy+Tdqdx-Vdydx~Q. These relations arecalled IntermediateIntegrals. Themethod ofprocedurewillbebestunderstood bystudying rorkedexamples. Ex.(i). 2x2r-5xys+2y*t+2(px+qy)=0. Proceedingasabove, weobtain thesimultaneous equations 2x*dy* +5xydydx +2y2dx2=*0, ........................(1) nd 2x2dpdy+2y2dqdx+2(px+qy)dydx^Q................... (2) (1)gives (xdy+2ydx)(2xdy+ydx)=0, i.e.xzyaorxy2=b. Ifwetakex2y=aanddivide eachterm of(2)byxdyoritsequivalent -2ydx, weget2xdp-ydq +2pdx-qdy=0 9 i.e.2px-qy=*c. This, inconjunctionwithx2 !/=a,suggeststheintermediateintegral 2px-qy= <f>(x2 y),.............................. (3) here isanarbitraryfunction. [Cf.Ex.(i)ofArt. 152.] Similarly xy2=6andequation (2)leads to px-2qy=\{,(xy2 )............................... (4) Solving (3)and(4), 182 DIFFERENTIAL EQUATIONS so i.e.z~ij<f>(x2 y).dlog(x*y)-iJWxy*).dlog(xy*), or z=(x*y) Ex.(ii). Eliminatingrand tasbefore, weareledtothesimultaneous equa- fcinsy*dy*+2ydydx +dxz=Q,........................ (5) and y*dpdy +dqdx-(p +6y)dydx=...................(6) (5)gives i.e. Usingthisintegral and dividingeachterm of(6)byydyorits equivalent ~dx,weget .e.py Thissuggeststheintermediateintegral Aswehaveonlyoneintermediateintegral, wemustintegratethis byLagrange's method. Thesubsidiary equations are dxdydz ' Oneintegralis2x+y*=a.Usingthistofindanother, i.e.z-y*+y<}>(2x-fy2 )=b. Hence thegeneral integralis \J,{z-y3+y</>(2x+y2 ),2x+1/2 }-0, or z*** Ex(Hi). pt-qs^q*. Thesimultaneousequations are qdydx +pdx*=*Q,...........................(7) and pdqdx-q*dydx*=>Q ............................(8) (7)gives dx= orqdy+pdx(=dz)=Q, i.e.x=aor z=b. Ifdx=Q(8)reduces to0=0. Ifz=*b,qdy-pdxand(8)reduces to p i.e. giving~-+x~c= \ls(z)................................ (9) SECOND ANDHIGHER ORDERS 183 (9)maybeintegrated byLagrange's method, butashorter wayia fcorewrite itg^ \jr---TH,), giving yxz-I\fs(z)dz+F(x) Examples forsolution. (1)r- Icos2x+ptanx=0. (2)(x-y)(xr-xs-y8 +yt)- (3)(q+l)s=(p+l)t. (4)J- (5)Xy(t-r)+(x2-y*)(s-2)=py-qx. (6)(l-f^)2r-2(l-f-^ +g+j (7)Find asurfacesatisfying2x2r-5xys +2y*t+2(px-{-qy)=0 and touching thehyperbolic paraboloid z=*x*~y2alongitssection bythe planeyl. (8)Obtain theintegralof$2r-2^5-f p2t=0 intheform andshow that thisrepresentsasurface generated bystraightlines that areaJlparalleltoafixedplane. *155. Monge's method ofintegrating Rr-fSs +Tt+U(rt-s2)=V. Asbefore, thecoefficientsJ?,/S,T,C7,Varefunctions ofp,q, x,y,z. Theprocessofsolution fallsnaturallyintotwoparts: (i)theformation ofintermediateintegrals ; (ii)thefurtherintegrationoftheseintegrals. Forthesake ofclearness weshall consider these twopaits separately. 156.Formation ofintermediate integrals. AsinArt. 154, r(dp-sdy)/dx and t=(dq-sdx)/dy. Substitute forrand tin Rr+Ss+Tt+U(ft-s2 )=V, multiply upbydxanddy(toclear offractions), andweget Rdpdy+Tdqdx+Udpdq-Vdxdy sayN-sM-0. *Theremainder ofthischapter should beomitted onafirstreading. Thin axtension ofMonge'sideas isduetoAndr6 Marie Ampere,ofLyons (1775-1836), whose name hasbeengiventotheunit ofeleotrio ourivmt. 184 DIFFERENTIAL EQUATIONS Wenowtrytoobtain solutions ofthesimultaneous equations M=0, #=0. Sofarwehave imitated themethods employedinArt. 154,but wecannot now factoriseMaswedidbefore, onaccount ofthe presenceofthetermsUdpdx+Udqdy. Asthere isnohopeoffactorising MorNseparately,letustry tofactorise M+\N, where Xissomemultipliertobedetermined later. WritingMandNinfull,theexpressiontobefactorised is Rdy2+Tdx2-(S+\V)dxdy +Udpdx +Udqdy +\Rdpdy+\Tdqdx+\Udpdq. Asthere arenoterms indp*ordq2 ,dpcanonlyappearinone factor anddqintheother. Supposethefactors are Ady+Bdx+Cdp andEdy+Fdx+Gdq. Thenequatingcoefficients ofdy2 ,dx2 ,dpdq, Wemaytake A=R,E=l,B=kT,F^l/k, C=mU,G=\/m. Equatingthecoefficients oftheother fiveterms, weget kT+R/k=-(S+xF),.....................(1) \R/m=U, .................................(2) \T, ...............................(3) \R, ...............................(4) U..................................(5) From(5),m=k,and this satisfies(3). From(2)or(4),m=Xfi/l7. Hence, from(1), \2(RT+UV)+\US +U2=Q.............. ........(6) SoifXisaroot of(6),thefactorsrequiredare i.e.~(Udy+\Tdx+\Udp).~(\Rdy + We shall thereforetrytoobtainintegrals from the linear equations Udy+\Tdx+ \Udp=Q........................(7) and \Rdy +Udx+\Udq=Q 9........................(8) where Xsatisfies(6). SECOND ANDHIGHER ORDERS 185 The restoftheprocedurewillbebestunderstood fromworked examples. 157.Examples. Ex.(i). 2*+(rf-s)=l. Substituting R=T=Q, S=2,U~V=linequation (6)ofthelast article,* weget X2-f-2X 4-1=0, aquadraticwithequalroots-1and-1. WithX-1,equations (7)and(8)give dy-dp=0, dx-dq^0 9 ofwhich obviousintegralsare y-p=const. and x-q const. Combining these asinArt. 154,wegettheintermediateintegral y-p=f(x-q). Ex.(ii).r+3s+*+(rt-*1)-!. ThequadraticinXcomes to soX=>-1or- . WithX=-1,equations (7)and (8)give dy-dx-dp=0, -dy+dx-dqQ, ofwhich obviousintegrals are p+x-y=>const..................................(1) and q-x+yconst .................................. (2) Similarly X=-\leads to p+x-2y=const .................................. (3) and q-2x+y=>const ..................................(4) Inwhatpairsshallwecombine these fourintegrals? Consider againthesimultaneousequationsdenoted byM=0,N=0 inthelast article. Ifthese areboth satisfied, thenM-fX1Ar=andM+X2N=arealsoboth satisfied (where XxandX2aretheroots ofthe quadraticinX).Therefore oneofthelinear factors vanishes forX^Xj andone(obviouslytheother one, orelsedy=Q)forX=X2. That is,wecombine integrals (1)and(4),and also(2)and(3), givingthetwointermediate integrals and p+x-2y=*F(q-x *Wequote theresults ofthelast article tosavespace, butthestudent is advised towork eachexample from first principles, 186 DIFFERENTIAL EQUATIONS Ex.(iii). 2yr4-(px+qy)$+xt-xy(rt-*2 )-2-pq. ThequadraticinXcomes to \*xypq-Xzt/(ps +qy)+x*y*=>0, giving *=*ylPorz/j. Substitutingin(7)and(8)ofthelast article, weget,after alittle reduction,pdy-dx +ydp-0,..............................(5) 2ydy-px dx-xy dq=Q,.............................. (6) ~qydy +xdx-xydp-=Q,.............................. (7) and -2dy+qdx+xdq=*Q............................... (8) Combiningtheobviousintegralsof(5)and(8),weget But(6)and(7)arenon-integrable.Thismaybeseenfrom the waythatpandqoccur inthem. Thus, althoughthequadraticinXhas two different roots, wegetonlyoneintermediateintegral. Examples forsolution. Obtain anintermediateintegral (ortwo ifpossible)ofthefollowing: (1)3r+ts+t+(rt-s*)~l. (2)r+t-(rt-s*)=l. (3)2r+te*-(rt-s*)=2e*. (4)rt-s'+l-O. (5)3s+(rt-s2)~2. (6)'qxr+(x+y)$+pyt+xy(rt-s2)*=*l- pq. (7)(q2-l)zr- Zpqzs+(pa-1)zt+z*(rt-s*)=p*+q*-1. 158.Further integration ofintermediate integrals. Ex.(i).Consider theintermediateintegral obtained inEx.(i)of Art. 157,y-p^f(x-q). Wecanobtain a" complete" integral involving arbitrary constants a,6,cbyputting x-q==a And y-p=/(a)=&,say. Hence dz~>pdx+qdy=>(y- b)dx+(x~a)dy and z**xy-bx-ay +c. Anintegralofamoregeneral form canbeobtained bysupposing thearbitraryfunction /occurringintheintermediateintegraltobe linear, givingy-p-ro(s-9)+n. IntegratingthisbyLagrange's method, weget z=-xy+<f>(y+mx)-nx. Ifix.(ii).Consider thetwointermediateintegralsofEx.(ii),Art.157P and p+x-2y**F(q- SECOND ANDHIGHER ORDERS 18? Ifweattempt todealwith these simultaneousequationsaswedealt vriththesingle equationinEx.(i),weget Iftheterms ontheright-hand sideareconstants, wegettheabsurd result that x, t/,p,qare allconstants ! Butnowsupposethataand$arenotconstants, butparameters, capableofvariation. Solvingthefourequations, weget p~y-x+f(a), q~x-y +f}, giving dz*pdx +qdy dy)+f(a) dx+/3dy Toobtain aresult freefromsymbolsofintegration, put (Za*(o) and f/(a)(Za*0 Now [jSF'dS) d/3=/3F(/3)-\F (/3)dj3 9integrating byparts, Hence --*(*- y)1-^(a)- r2=-i(x-t/)2 -</> orfinally-jx=/3-a, These threeequationsconstitute theparametric form oftheequation ofasurface. Asthesolution contains twoarbitrary functions,itmay beregarded asofthemostgeneral formpossible. Examples forsolution (completingthesolution ofthepreceding set). Integrate bythemethods explained above : (1)p+a;-2y-/te-23 +3y). (2)p-x-f(q-y). (3)p-e*=f(q-2y). (4)p-y (5)p-y~f(q-2x), (6)px-y~f(qy-x). p-2y-F(q-x). (7)(zp-x)=f(zq-y). (8)Obtain aparticularsolution of(4)byputting <f>(a)-}a2 , ^(B)="iy82andeliminating aandp. 188 DIFFERENTIAL EQUATIONS MISCELLANEOUS EXAMPLES ONCHAPTER XIV. (1)r-2i/a . (2)log- +y. (3)2yq+y*t (4)r-2s+*=sin (2x+3t/). (5) (6)rx*-35X2/4- 2ty2 (1) (8) (9) (10)rtf-s2-s(sinx+siny)=sin a;siny. (11)7r-8s-3* +(rt-52H36. (12)Find asurfacesatisfyingr=6x-f2 andtouchingz=*x*+y* alongitssection bytheplanex+y+1=0. (13)Find asurfacesatisfying r-2s+J=6andtouchingthehyper- bolic paraboloidz=xyalongitssection bytheplane y=x. (14)Asurface isdrawn satisfyingr+=andtouchingx2+s2=l alongitssection by 2/=0.Obtain itsequationintheform z2(*2+z2-l)=2/2(z2+z2 ). [London.] (15)Show that ofthefour linear differentialequationsinx,y,ptq obtained bytheapplicationofMonge's method to 2r+qs+xt-x(rt-s2 )=2, twoareintegrable, leadingtotheintermediateintegral while theother two,although non-integrable singly, canbecombined togivetheintegral p-fJ^2-x=a. Hence obtain thesolutions z=\x*-2mxy-mV+nx4-(y+jmx1 ) and z-(a-J&2 )x4-Jx2+by+c, andshow thatone isaparticularcase oftheother. (16)Asurface issuch that itssection byanyplane paralleltox=0 isacirclepassing throughtheaxis ofx.Prove that itsatisfies the functional anddifferentialequations (17)Obtain thesolution ofx2r+2xys+y2t~0 intheform andshow that thisrepresentsasurfacegenerated bylinesthat intersect theaxis ofz. (18)Show thatrt-8*=*Q leads tothe" complete" integral MISCELLANEOUS EXAMPLES 189 Show thatthe" general" integral derived from this (asinArt. 134) representsadevelopable surface (seeSmith's Solid Geometry,Arts. 222-223). Hence show that foranydevelopablesurfaceqssef(p) (19)Find thedevelopablesurfaces thatsatisfy pq(r- 1)-(p2-g)s+(py-qx)(rt-*a )0. [Assume Q~f(p). This iscalled Poisson's method. Weget q*=ap or7>2+2=&a , giving **<p(x+ay)orz=bxcosa+bysina+c. Thesecond oftheseintegrals representsaplane whichgeneratesthe developablesurfacegivenbythecorresponding" general" integral.] (20)Show that if then r-TI(RT-S*) y--~S/(RT-S2 ),t-R/(RT-S*) t ^7 where 72= ^-^,etc. Hence show thattheequation ar+&s+ct+e(rt-s*)=*0 transforms into AT-BS+CR+E-0, where a,6,c,eareanyfunctions ofx,y,p,q,andA,B,C,Ethecorre- spondingfunctions ofP,Q,X,F. Applythis Principle ofDuality (cf.No.21oftheMiscellaneous ExamplesattheendofChap. XII.)toderive twointermediateintegrals ofpq(r-t)-(p2-q*)s+(py-qx)(rt~s2 )=0. (21)Prove that ifx,t/,u,varerealandu+iv=f(x +iy),thenFti andV=vareboth solutions of andthetwosystemsofcurves w=const., v=const., aremutually orthogonal. Verify thesepropertiesfortheparticularcases (i)u+tt;=sc+t'y, (ii) (iii) [Thedifferential equationisthetwo-dimensional form ofLaplace's equation,which isoffundamental importanceingravitation,electro- statics andhydrodynamics, uandvarecalled Conjugate Functions. SeeRamsey's Hydro- Mechanics, Vol. II.Art.41.] (22)Obtain thesolution of P.D.B. l&O DIFFERENTIAL EQUATIONS subjecttotheconditions y*=f(x) andJ~**F(x) when t=0,intheform 1Cx+at s- F(\)d\.*aJx-at [yisthetransverse displacementofanypointa;ofavibrating stringofinfinitelength, whose initial displacement and velocityare given byf(x)and F(x). SeeRamsey's Hydro-Mechanics,Vol. II. Art. 248.] (23)Ify=f(x) cos(nt+a)isasolution of show thatf(x)**Asinmx+Bcosmx+Hsinhmx+Kcoshmx,where w=\/(n/aa ). [The differential equationisthatapproximatelysatisfied bythe lateral vibrations ofbars, neglecting rotatoryinertia. SeeRayleigh's Sound, Art.163.] (24)Show that w=*A sin(rmrx/a)sin(mry/b)cos(pet+d) ,. d*w9fd2wd2 satisfies ._..(__+ andvanishes when 2=0, ?/=0, #=aory=5, provided thatmandnarepositive integers satisfying (p/ir)-(m/a) +(n/6). [This gives onesolution ofthedifferential equationofavibrating membrane with afixedrectangular boundary. SeeRayleigh's Sound, Arts. 194-199.] (25)Show that w=AJ(nr)cos(net+a) ,.n o1div\satisfies-cITl+" "a" )d^2\3/*2ror/ whereJisBessel's function oforder zero (seeEx.2ofthesetfollowing Art. 97). [This refers toavibrating membrane withafixed circularboundary. SeeRayleigh's Sound, Arts.200-206.] (26)Show thatV=(Arn+Br~n-1 )Pn(cos 0) ..fid*V2dV 192FcotOBVAsatisfies ^H---^^-^+-_ ^-=0, wherePnisLegendre's function oforder n(forLegendre*s equation, BeeEx.2ofthesetfollowingArt.99). [2V.B.Take/x=cos0 asanew variable. This equationisthe formtaken byLaplace's potential equationinthree dimensions, when Visknown tobesymmetrical about anaxis. SeeRouth's Analytical Statics, Vol. II.Art. 300.] CHAPTER XV MISCELLANEOUS METHODS 159.Thischapterconsists ofsixsections. The first(Arts.160- 161)issupplementarytoChap. VL,anddeals withsome difficulties inthetheoryofsingular solutions, especiallythedefinition ofan envelope andthewayinwhichparticularsolutions mayoccur in thediscriminants. Theconceptionofdiscriminant-loci asboundaries appearstobeverylittleknown. Thesecond section(Arts. 162-167)deals with Riccati'sequation, chieflyinitsgeneralisedform. Theexamplesinclude aseries which indicate inwhat cases Eiccati'soriginal equationcanbeintegrated infinite terms. The third section (Arts. 168-170)deals with total differential equations, and issupplementarytoChap. XI. Theuseofan integratingfactor forhomogeneous equationswillappealtothe elementary student, whileMayer's method isofgreatinterest from thepointofview oftheory. Thefourth section(Arts. 171-177)deals with linear differential equationsofthesecond order and their solution byseries. Itis supplementarytoChaps.IX.andX.Afew results concerning equationsofhigherorder areincluded. The fifth section(Arts. 178-181)deals withsomeequationsof MathematicalPhysics, particularlythose concerned with wave- motion. ItissupplementarytoChaps.IV.andXIV. Finallythesixth section(Arts. 182-183)deals with numerical approximationstothesolution ofdifferential equations (supple- mentarytoChap. VIII.). Afterdescribingthemethod ofAdams, perhapsthebestthathasyetbeen devised, itgivesasummaryof some extensions (due toE.Remes)oftheauthor's method(i.e.that ofArts. 90-93). 192 DIFFERENTIAL EQUATIONS 160.Some difficulties inthetheory ofsingular solutions.* We shallnowsupplement Chap.VI.bypointingoutsome diffi- cultiesconcerning envelopes, singular solutions, andparticular integrals. Theolddefinition ofanenvelopeofafamilyofcurves, asthe locusoftheultimate intersectionsofconsecutive curves, must be abandoned, forithasbeenfound toleadtotheridiculous conclusion thatacurve isnottheenvelopeofitsown circles ofcurvature. fDe laVallee Poussin's definition isthelocusoftheisolated characteristic points (i.e.ofordinary points onacurve whose distances fromneigh- bouringcurves aresmall toanorder beyond thefirst). However, ithasbeenpointedoutthat this isstillunsatisfactoryincertain respects. fForourpurposesthemost convenient definition appears tobeacurve which toucheseverymemberofthefamily,andwhich, at eachpoint,istouchedbysomememberofthefamily.Thisagreeswith thedefinitiongivenonp.G6;thesecondpartofthedefinition was notexplicitlystated there, but itwasimplied bythefollowing sentence. There areatleast three different definitions ofasingularsolution. Our definition(p.66)isthat itisasolutioncorrespondingtoan envelope ofthefamily ofcurvesrepresented bythecomplete primitive. However, inexceptionalcases theenvelopeisalsoaparticularcurve ofthefamily. Thus theparabola y~c(z-c)2touches theliney~Q atthepoint (c,0),soy= istheenvelopeofthefamilyobtained by givingallpossiblenon-zero values toc,aswell astheparticular curvegivenbyc=0. Inaccordance withourdefinition, ?/~0must beconsidered tobebothasingularsolution andaparticular integral ofthedifferentialequationofthefamily (Ex. 6,p.76).Butsome prefertoconfine thetermsingulartoasolution which cannot be obtained bygiving anyconstant value tothearbitraryconstant occurring *Forenvelopes, seeFowler's Elementary Differential Geometry ofPlane Curves, Chap. V.Forsingular solutions, seetheEncyklopddiederMathematischen Wissen- tchaften U.A4aand III.D8. fCand C",thecentres ofcurvaturecorrespondingtotwoneighbouring pointsPandP'ofacurve, lieontheevolute ofthat curve. Thedifference between the radii ofcurvature CPandC'P* isthearcOC1oftheevolute. This arc isingeneral greater than thechordCO7 ,i.e.greater than thedistance between thecentres of curvature. Thus onecircle ofcurvature completely encloses theother, andthere arcnorealintersections. Forother caseswhere theolddefinition fails, seeEx. 13, following Art. 161. JNeville, Proc.Camb. Phil. Soc.tVol.XXI.p.97,1922. But seetheendofthis article fortheexceptionalcase ofenvelopes parallel totheaxisofy. SINGULAR SOLUTIONS 193 inthecomplete primitive. Athird definition*ofasingularsolution isthat itisasolution which occurs inthep-discriminant.Itwillbe shown inArt.161thatsuchasolution neednotrepresent anenvelope. Itmaybeaparticular solution, oritslimiting form. Itisnatural fqrthestudent tosupposethatevery familyof curvesdepending ononeparameterwillpossess anenvelope and consequentlythateverydifferential equationofthe firstorderand ofdegree higher thanthe first willpossessasingularsolution. But this isnotthecase. Indiscussing envelopes,itisimplicitly assumed that thefunctionsoccurringintheequationofthefamily satisfy certain conditionsconcerning continuity. These conditions are usuallysatisfied forthecomplete primitivesofthesimpledifferential equations giveninanelementary treatment ofsingular solutions, but this isduetothefactthat inconstructing suchexamples the complete primitives werereally taken asthestarting point.If westartfrom themostgeneraldifferential equationofsimilar form, there isnoreason tosuppose thatthecomplete primitivewillsatisfy theconditionsrequiredfortheexistence ofanenvelope. Infact, wemaysaythattheexistence ofasingularsolution must becon- sidered astheexceptionrather than therule.f Itshould benoticed thattheusualprocessforfinding envelopes (Art. 56)mayfailforoneform ofthecomplete primitive,andyetbe effective foranother. Forexample,itfails for#*+y*=c%orfor x+sin~12/=c, but iseffective for (x+y- c)2=&xy,orfory=sin(c-x). Theequationx^+y^ c^,leadingtoy~xp2 ,illustrates another point. The differentialequationissatisfied byy=0, buthardly byz=0,which, giving p=co,leaves both sides indeterminate. However, x=0andyQarebothenvelopesofthefamilyofcurves (parabolas touchingtheaxes) andbothsatisfy y(dx)2~x(dy)2 ,a differential relation whichreally representsthegeometricalfacts moreaccurately than thedifferentialequation. [Cf.Ex. 9,p.79 andEx.11,p.233. Inthe firstx=0 isalimiting form ofapartic- ular curve, and inthesecond anenvelope andalsoacusp locus.] Insuch caseswefeelcompelledtorefusex=0 aplaceamong *This istheoneadopted bymostadvanced treatises (cf.Ince's Ordinary Differ- ential Equations, p.87,andBieberbach'sDifferentialgleichungen, p.85). Inquoting results from various sources itisnecessary togivethedefinitions onwhich theyare based, ormuch confusion maybecaused. fSeeEx. 10,followingArt. 161. 194 DIFFERENTIAL EQUATIONS thesolutions, buttherejection maybeconsidered asduetothe failure ofthe differentialequationtorepresent fairlydirections paralleltotheaxis ofy,rather than toany peculiarityinthe envelopeitself. 181. Discriminants, Particular Solutions, andBoundaries. Inthis article weshall confine ourselves tocomplete primitivesofthe form/(x, y,c)=0, where/(#, y,c)isapolynomialinx,ytandc,which mayalsobewritten intheform ofoy}cn+nal(x,y)cn~l+Jn(n- I)a2(z,y)cn~24-...+an(x,y)=0. Thec-discriminant Acisdefined(exceptforanumerical factor) as theproductofa2n~2andthesquaresofthedifferences oftheroots. Thea2n~2isintroduced tomake theresult apolynomialinaor aj...an.Thus forn=2, 3,4wegetrespectively 2\ffjn_/72\ AsinChap.VI.weshallsometimes usetheword discriminant to denote, notonlythefunction Ac,butalsotheequation Ac=:0andthe locirepresented bythisequation. Inworking examples onsingularsolutions itisdesirable to employ.asystematic method ofcalculatingthediscriminants. For quadratics, cubics andquartics,theabove results may beused.* If,asinArt. 56,weobtainAcbyelimination, there isariskthatsome factors willbeoverlooked. Itisoftenrecommended thatSylvester's dialytic method should beused toperformthis elimination. To applythis here,wemultiply /bycn~2 ,cn~3 ,...c,1,anddf/dcby c"-1 ,cn~2 ,...c,1,andthen eliminate c2n~2 ,c2"-3 ,...c,1from the (2n-1)equations thusformed, givingadeterminant of(2n-1)rows andcolumns. Forthequadraticac2-h2a1c+a2^0, thisgives a,2al9a2 2a,2aj, =4a(aa2-a12 ). 0, 2a,2aj Butthiscontains thesuperfluousfactor a .Itiseasytoseethatthe samesuperfluousfactor willoccur whatever thedegree of/, giving anexpressionofdegree (2n-1)instead oftheproper degree (2n~2). IfSylvester's method isemployedfortheexamplesattheendofthis article, thisfactor must beremoved. *Inusing these, remember thatthea'sarenottheactual coefficients, which have alsobinomial numerical factors;e.g.foraquartic thecoefficient ofc2isnota,,but 60,. SINGULAR SOLUTIONS 195 Theprimary purposeoftheseexamplesistoillustrate someways inwhichparticularsolutions ortheirlimitingformsmaybegiven bythe c-and*^-discriminants.Insome cases thesolutions occur asmerelyonepartofaparticularcurve (Ex. 1).Theirgeometrical significancetakes various forms. Theymaybeenvelopes andso alsosingularsolutions (Ex. 2),ornode-loci (Ex. 3),orcusp-loci (Ex. 4),ortac-loci(Ex. 5),orasymptotes (Ex. 6),ortangents touchingallthecurves ofafamilyatthesamepoint (Ex. 8).They maybemerelylines (nottangents) throughacommonpointofa family (Ex. 7).Inconnection with Clairaut's formtheyare furnished (Ex. 9)bytheinflexionaltangentstotheenvelope. Itissometimes stated thatwhenparticularsolutions occur in thediscriminants, theydosotothe firstpowerinAc,andcubed inAp.This rulemaybecombined with those ofArt.64inthe symbolical form:AC-A72C3P,Ap=AT26fP8 ,where E,N,C,P,T denoteenvelope, node-locus, cusp-locus, particular solution, and tac-locusrespectively.These rules areuseful assuggestionsin simple cases, butexamplesinwhichtheyfailareeasilyconstructed (Exs. 3,4,6,13,14). Weshallnowexplaintheconceptionofparticularsolutions and otherexceptionallociasboundaries.* Werestrict ourselves tothe casewhere/(x, y,c)isapolynomialinx,y,c,andsuch that corre- spondingtoevery pairofrealvalues ofx,ywegetanequationinc ofdegree nwith, say,mrealrootscorrespondingtorealcurves, and (n-m) imaginaryrootscorrespondingtoimaginarycurves. We furtherstipulatethat theroots, which are, ofcourse, functions of xandy,shallvary continuously when xandydoso. Letacertain curveB(x,y)=(notoccurringinamultiple form, ormade upofanumber ofsimpler curves) beaboundarybetween tworegions,inoneofwhichmhasacertain valueMandintheother avalueM-2.Asthepoint (x,y)travelscontinuouslyoutofthe firstregion,across theboundary B,intothesecond, apairofreal unequalrootsbecome lessunequal,thenequal (onB)andfinally (inthesecondregion) conjugate complex. Ac,which contains the squareofthedifference ofthese roots, must vanish onBandthen change sign,asthesquareofthedifference oftwoconjugate complex *Hereandelsewhere Ihavemade considerable useofsome valuable suggestion* made byMr.H.B.Mitchell, formerly Professor ofMathematics atColumbia Uni- versity, NewYork. However, hemust notbeheldresponsibleformytreatment, forourpointsofview arerather different. 196 DIFFERENTIAL EQUATIONS roots isnegative. B(x, y)must alsochange signas(x,y)travels across it.Moregenerally,ifmchangesfromMtoM2r,where r isanoddinteger, Acwillchange sign,andB(xty)willoccur inA toanoddpower (which, however, neednotber;cf.Ex.14,where B(x, y)occurs cubed, butr=l).Ifrisaneveninteger, B(x 9y) occurs toanevenpower. Conversely,ifB(x, y)occurs toanodd power,rmust beodd. However,ifB(x,y)occurs toanevenpower, sothatAcdoesnotchange sign,rneednotbeeven;itmaybezero, asinEx. 13,whereBisanenvelope which iscrossed byallthe curves ofthefamily. Insuch cases theenvelopemust occur toan evenpower, contrarytotherule&C=EN2C*P. Similar considera- tionsapplytoAp,onreplacingthenumber ofrealcurves through apoint bythenumber ofreal directionsthroughit.Aspecially interestingcase isthat ofClairaut's form(Ex. 9).Aninflexional tangenttotheenvelope correspondstotwoequalroots inp,andso leads toAp=0.AsforClairaut's formAc=A^,Ac also. Analternativegeometrical method*ofinvestigating singular solutions istoreplace pbyz,thusconvertingthedifferentialequation intothealgebraic equationofasurface.Similarly,inthecomplete primitivecmaybereplaced byz.Thismethodrequiresagood knowledgeofthegeometryofsurfaces. The difficulties inthetheoryofsingularsolutions aregreateven fordifferentialequationswith coefficients which arepolynomialsin xandy.When thecoefficients aretranscendental functions, with singularitiesofvariousdegreesofcomplexity,the difficulties are greatly increased.! Examples forsolution. [Weshall useC.P., Diff.Eq.,Ac,A^,and S.S., todenoterespectively complete primitive,differentialequation, c-discriminant, p-discriminant, andsingularsolution. AcandAphavebeenobtained from theformulae given above, butnumerical factors havebeenomitted. Thestudent should draw rough graphs (without calculating exact values ofxandy)which willshow theform ofafewmembers ofeach familyofcurves and theirpositionrelative tothe locigiven bythe discriminants.] (1)Given theC.P.y(x+c)+c2=0,obtain theDiff.Eq. also Ac=y(4:c-t/), *EncyklopadiederMathematischen Wissenschaften,III.D8,orQoursat'i Conn?Analyst Mathtmatique, Vol. II.4thed.,Art. 435. tM.J.M.Hill, Proe. Loud. Math. Soc.tSeries 2,Vol. 17,1918, p.149. SINGULAR SOLUTIONS 197 [The O.P., fornon-zero values ofc,representsafamilyofrectangular hyperbolas, y=0isanasymptoteofallthesehyperbolas, andalsopart oftheparticular integral zt/=0 obtained from theC.P.byputting c=0. y4zisanenvelope (aS.S.). The rules&C~EN2C*P,APT2CP3 holdgood. Theplanecanbedivided intofourregions,intwoofwhich thenumber ofrealcurves ofthefamily through anypointistwo,while intheother tworegionsthenumber iszero. Theboundaries between theseregionsarethelocigiven bythediscriminants, andbothoccur to oddpowers. Thisagreeswithourtheoryofboundaries, forinthiscaseM=2,M-2r=*0, sor-1,which isodd.] (2)Given theC.P.t/-c(z-c)2 ,obtain theDifi.Eq. p3-4xyp-f8y2=0, alsoAfl-y(27y-4x), Ap-^(270-4^). [Asmentioned inArt. 160,y=*0isanenvelope (aS.S.)andalsoa particular integral. Moreover, itmay beregardedasatac-locus. 27y=4x3isanenvelope. Thesecond andfourth ofthesegeometrical interpretations, butnotthe firstandthird, aresuggested bytherules (3)Given theC.P.4y2=3c2z(z-c)2 ,obtain theDiff.Eq. alsoAc [The calculation ofthediscriminants israther laborious. y=0isa node-locus aswell asaparticularsolution. cc= isacommon tangent attheorigintoallthecurvesexceptthat forwhich c=0.(Of.Ex.8.) 3z5=64t/2istheenvelope. Tounderstand whythevarious factors in thediscriminants occur tooddorevenpowers wenotice thatxQisa boundary betweenregionswhere thenumber ofrealcurvesthrough any pointincreases from zero totwo, while theenvelopeistheboundary betweenregionswhere thisnumber increases fromtwotofour. On y=thefour coincide inpairs,butoneach side ofthepositive part, between itandabranch oftheenvelope,thenumber isthesame, namely, four. The rulesAC-^]V2C3P,Ap-M^CP3fail tosuggestthe geometrical interpretationofthelocias=0,andy=0.] (4)Given theC.P.4y3=c(3a?-c)2 ,obtain theDifi.Eq. also Ac [The C.P., fornon-zero values ofc,representsafamilyofsemi- cubical parabolaswithcusps ony0,which isacusp-locus and also aparticularsolution. y*=x*isanenvelope (aS.S.). The rules AC=##2C3P,AP=#T2CP3suggestthatt/=0isacusp-locus,butthey failtoindicate that itisalsoaparticular solution.] (5)Given theC.P.y2=c(3a;-c2 ),obtain theDifi.Eq. also Ac-y4-4ar>,Ap-yV-4s3 ). 198 DIFFERENTIAL EQUATIONS [The C.P., fornon-zero values ofc,representsafamilyofparabolas withy=0asaxis,anypointofwhich isthevertex oftwosuchparabolas with their concavities turned opposite ways. y=0isatac-locus and alsoaparticularsolution, y4=4a^isanenvelope (aS.S.). y2=c(3x-c2 ) touches theenvelopeatthepoints {c2 ,>/(2c3 )},which areimaginary ifcisnegative, andintersects itat{Jc2 ,W(-c)}>which areimaginary ifcispositive. Therulessuggestthetac-locus, butnottheparticular solution.] (6)Show that forallvalues ofmexcept 0,thecomplete primitive Ofytn-2^2Ijg4^fnsm2(x+c)2 Show that forthethree cases,manoddpositive integer greaterthan 1, m=1,andmanoddnegative integer, AcandAParerespectively and tf~\ y,y2-, providedthat these discriminants areobtained fromequationsmulti- pliedbytheleastpowerofynecessarytogetridofnegative powers. [y= isinthe firstcaseacusp-locus,inthesecond anenvelope (a S.S.),andinthethird thelimiting form ofaparticular solution, which is asymptotictoallcurves included inthecomplete primitive.c=oo gives t/~m=0, ifmisnegative,soingeneralthislimiting form ofa particular integral contains thesolutiony=Qinamultipleform. If m= 1,butnototherwise, theparticularsolution occurs tothepowers given bytherulesAC=W2C3P,Ap=r2CP3 .The rulesgivethe powersofthecusp-locus correctly onlyform=3.] (7)Given theC.P.y=x(x+c)2 ,obtain theDifl.Eq. x2p2-2xyp+y2-&x3y=0, also Ae-zy, Ap=xb y. Show thati/=0isanenvelope (aS.S.),and#alimiting form ofa particular solution, butnot itself asolution. [The vanishingofthediscriminants attheorigin,apointcommon toallthecurves ofthefamily,could have beenpredicted. Forsince attheorigintheequationofthefamilyissatisfied foranyvalue ofc, thecoefficients ofevery powerofcandalsothetermindependentofc vanish there, henceAc=0,forevery term initvanishes. Asthecurves have different tangentsatthecommonpoint,theDifi.Eq.issatisfied there foranyvalue ofp,sobyanargumentsimilar tothat for AC,AP=0. (Cf.Ex. 7,p.79).] (8)Show that forallnon-zero values ofc,thecurves ofthefamily y*=x(x+c)2touch sc=0 attheorigin. Obtain the Difi.Eq. ix2p2-ixyp+y2-x*0, alsoAc=xy2 ,Ap^ar5 . Show that2/=*0isanode-locus, while =0 isalimiting form ofa particularsolution (though notitself asolution), andalsoalinetouching allthecurves, exceptthat forwhichc0, atonepoint. (Suchaline doesnotsatisfyourdefinition ofanenvelope.) SINGULAR SOLUTIONS 199 [As inEx. 7,Acmust vanish attheorigin. Apalsovanishes (althoughthecurves thistimehave notdifferent tangents).Cf.Ex. 9, p.79.] (9)Show that forthedifferentialequation (ofClairaut's form) [27t/=4z3istheenvelope (aS.S.) ;t/2=0 isaparticular solution, andrepresentstheinflexional tangenttotheenvelope. Nowthrough anypointthree tangentsto27y=4x3canbedrawn. Allofthese are realfortheregioninthefirstquadrant between thecurve andy=0,also forthesimilarregioninthethirdquadrant. Fortheotherregions two areimaginary. Forapoint ony=0twoarecoincident, sot/=0must occur inthediscriminants. Similarly,\\henever theenvelopesolution ofanyother differential equationofClairaut's formpossessesinflexional tangents,these occur inthediscriminants.] (10)Given adifferentialequation /(*,y,j>)-o ................................... (i) deduce that +.p+ =o.................................. (2)3xr ^>ydx^p Hence show that foranypointonasolution given bythe^-discrimi- nant, forwhich +-o........................................... (4)dxrdy Equations (1), (3)and(4)arenecessaryconditions forasingular solution. ForClairaut's form/(#, y,p)~y-px-F(p) >soequation (4) issatisfiedidentically. But ingeneralthere isnoreason whyallthree shouldpossessasimultaneous solution, soingeneraladifferential equationhasnosingularsolution. [ApplyingthistoEx.(i)onp.75,wefindthethree conditions are l-t/=0, giving j>=0,satisfies allthree, but2-3?/=0 does not satisfythefirst.] (11) [Inthisexamplethethird definition ofasingular solution (Art. 160)istobeused. Ex.10holds forallthreedefinitions.] Show that ifacurve exists forevery pointofwhich thethree equations have acommon solution inX,then alongit 200 DIFFERENTIAL EQUATIONS ofandhence dy' ety Hence show that if- ^/=0,\**pandthecurve isasingularsolution y 7-^f T^f ofthedifferentialequation /(cc, y,p)=0,while if^~=*0,then =0also. oy ox [This shows that thenecessary conditions forasingular solution, giveninEx. 10,become sufficient bytheaddition ofthecondition 7}f "^f-=0.Butthislastcondition isnotnecessary.InEx.2,=16y-4zp. O?/ ot/ This iszero foroneenvelope y=0,butnotfortheother, 27y=4z3 .] (12)Show that thelocus ofthepointsofinflexion ofthecurves represented bythecomplete primitiveofequation (1)ofEx.10satisfies equation (4)ofEx.10,andhence willbeincluded intheresult obtained byeliminating pbetween theseequations. ApplythisprocesstotheequationsofEx. 7,performingtheelimina- tionbySylvester's method, andobtain6y(4y~se3)=0. [Notice that allthelociofApareincluded, aswellasthelocus ofinflexions4?/=a;3 .] (13)Show that theequations y2=(x-c)3 ,?/=(z-c)3 ,a+y$=A allrepresentfamilies ofcurves inwhich neighbouringcurves donot intersect inrealpoints, andyetanenvelope yexists. (Inthethird casex0 isalsoanenvelope.) Obtain thecorrespondingDifE.Eqs.,Sp3 27t/, ?)3=27y2 ,xp*+y0 ; c-discrirainants, y4 ,y2 ,x*y*(x-y)2(x+y)*; and' p-discriminants, t/2 ,y4 ,x2 t/2 . [Notice that inallthese cases theenvelopeoccurs toanevenpower, forthereasongiveninthediscussion ofdiscriminant-loci asboundaries. Forthe firstandthird families theenvelopeisalsoacusp-locus,so ordinaryrules hold, butthis isnotsoforthesecondfamily. The loci x-t/=:0, x+yQarcwhere twoimaginary curves, given bynegative values ofcintheequationofthethirdfamily, become coincident.] (14)Show thatt/=(a?-c)4representsafamilyofcurves havingfour- pointcontact with itsenvelope y=0. Obtain thecorrespondingDifi. Eq.p4=256i/3 ,anddiscriminants A<= 2/3 ,AP=y9 - [The envelope againoccurs toapower higherthan the first. This time thepowerisodd, asitshould be,since thenumber ofrealcurves through anypointistwoononeside oftheenvelope, andzeroonthe otherside.] (15)Show that each oftheequations x*+y*=c*,x*+y*c, (x+y-c)2=4zy, (x+y-c2 )2=4xy, representsafamilyofparabolas with acommon axisbisectingtheangle xOy,andhavingx=0andt/=*0 asenvelopes. Show thattheattempttodetermine Acfails forthe first andsecond forms (oritmaybeconsidered togive 1,theequation ofthelineatinfinity, which touches allparabolas),while forthethird Axy9andforthefourthAx2y2(x-y)2 . RICCATI'S EQUATION 201 [*-y0isaparticularcurvecorrespondingtoc0.Indiscussing discriminants weshould avoid forms likethe firstandsecond, inwhich theterms arenotsingle- valued, andalso likethefourth, where different curvescorrespond todifferent values ofc2andnotofcitself.] 162. Riccati's equation. Thisname wasoriginally giventothe differentialequation* where6,c,andmareconstants. Foracertain setofparticular values ofmitcanbeintegratedinfinite terms(seeExs. 7-14below), butingeneralthesolutionrequiresinfinite seriesclosely connected with BesselFunctions.f ByaRiccati'sequationisnowusually understood thegeneralised form y^P+Qy+fijf ,..............................(1) where P,Q,andRarefunctions ofx.Thisequationisofsome importanceinDifferential Geometry.^ 163.Reduction toalinear equation ofthesecond order. Put When wesubstitute inequation (1)theterms inudisappear. Hence, onmultiplying upbyR2uyweobtain i.e.Riit-(QR+#>! +PR*u=0, ..................(2) alinearequationofthesecond order. Inspecialcases(asinthe examples below)thismaybeintegratedinfinite terms, but in generalsolution inseries willberequired. However, ineverycase thesolution willbeoftheform u=Af(x)+BF(z) 9 mvinor 41_i___ sr_giving y- citf(x)+RF(x)' whereA/Bhasbeenreplaced byc. *Suffixes denote differentiations with respect to*. fForthehistory ofRiccati's equation and itsconnection with Besee) Functions, leeWatson's Theory ofBeaad Functions, pp.1-3and85-94. JThere are20references toRiccati intheindex ofDarboux's Lemons surla Theorie Generate desSurfaces. SeealsoEisenhart's Differential Geometry, pp.25, 168,249,429,andForsyth's Differential Geometry, pp.20,383. This propertyistherealreason forchoosingthesubstitution andenables usto recall itifitisforgotten. 202 DIFFERENTIAL EQUATIONS Thisgivestheimportantresult that thegeneral integral ofRiccati's equationisahvmographic function oftheconstantofintegration. Conversely,itiseasily shown(asoutlined inEx.6below) thatwe obtain aRiccati'sequation byeliminatingthearbitraryconstant c fromanyequationoftheform cg(x)+G(x) 164.The cross-ratio ofanyfour particular integrals ofaRiccati's equation isindependent ofx.Wemaytakethefourintegralstobe p(x), q(x), r(x), s(x),which arederived from+j/b7givingc thefourspec/alvaluesa,/8,y,S. Then o-ag+G ^+G_(a-l3)(gF-f6)Jhen-_---_-t with similarexpressionsfortheother differences ofanytwo of p9qyrys.When weform thecross-ratio,allthefactors involving functions ofxcancel out,andweobtain whereCisindependentofx. 165.Method ofsolution when three particular integrals areknown. Letthese beq(x), r(x), s(x). Then itfollows from thelastresult, withp(x)replaced byyythatthegeneralsolution is {y-q(x)}{r(x)-s(x}}_ sointhis case thegeneralsolution hasbeen obtained without quadratures (i.e.,withoutintegrations), 166.Method ofsolution when twoparticular integrals axeknown. Letthese beq(x), r(x). Then, asyi and Similarly y^r^y- r){Q+(y+ r)R}. Hence l^i_2^=(?_r)y-q y-rw giving log^3-=c4-\(q-r)Rdx, y * BOinthiscasethegeneralsolutionrequiresonequadrature- RICCATI'S EQUATION 203 167.Method ofsolution when oneparticular integral isknown. Let fchisbeq(x). Thesubstitution*y~q(x) +-transforms equation (1)into But, sinceq(x)isanintegral, Subtracting andmultiplying upbyz2 ,weget or zl+(Q-{-2qR)z=-R9 alinearequationwhich canbesolved bytheuseofanintegrating factor exp|\(Q+2qR)dx \.Thedetermination ofthisfactorrequires onequadrature andthecompletionofthesolution(asinArts. 18-20) requires another, making twoinall. Examples forsolution. InExs. 1-5thestudent should work from firstprinciples, imitating themethods usedabove. Heshould notmerely quotetheresults and substitute inthem. (1)Byreduction toalinearequationshow thatthesolution of is (2)Show thatthesolution of yfyl+2- is (3)Show that tanxisoneintegralofyl=1-ft/2 ,andhence obtain thegeneralsolution intheform t/(c-tan x)ctanx+l. (4)Show that there aretwovalues oftheconstant kforwhich kjxisanintegralofx2 (yl+y2 )=2,andhence obtain thegeneralsolution. [&=2or-1;y(cz4-z)=2co;3+l.] (5)Show that1,x,x2arethreeintegralsof x(x*-l)y+x*-(x2-l)y- 1/2-0, andhence obtain thegeneralsolution *Thisappearsartificial. Amore natural (but longer) method isfirst toput yssq(jc) +u,which willgiveanequationofKiccati's formwithPreplaced byzero. Butthis isaspecial case ofBernoulli's equation (Art. 21),andtheusualmethod of solutionrequires thesubstitution l/u=z. Bycombiningthesetwosubstitutions wegetthatgiveninthetext.^ 204 DIFFERENTIAL EQUATIONS (6)Byeliminatingthearbitrary constant cfrom theequation cg(x)+G(x) y~cf(x)+F(x) obtain theRiccati'sequation (gf-Gf)yi=(gG^g.G) +(Gf,-QJ-gF^g^y +(fF l-fiPtf. (7)Show thatwhenm=*0 Riccati's equation canbeintegratedinfinite terms. \-l)^c(Ae^k-l) twherek**J(bc),ifbeispositive; yk*=ctan(A-kx), wherek^J( -6c),if6cisnegative; ycx+Atif6*0; l,ifc-0.] (8)Show thatthesubstitutionyz/ztransforms Riccati'sequation into andhence show thatthelatter equationcanbeintegratedinfinite terms ifw=0. [Use theresult ofEx.7.] (9)Bythesubstitutionz**yx* 9transform theequation xz1-az+bz*=cxn Into &1-y1+6ya-aul|-aB . Bythefurther substitution J=xobtain anequationofRiccati's form, -with b,c,mreplaced by6/a, c/a,(n-2a)/a respectively. Hence show thatthe firstequationofthisexamplecanbeintegratedinfinite terms ifn=2a. (10)Show that thesubstitution z=7-1- transforms the firstv 'bu equationofEx.9intoone ofsimilar form with a,6,creplaced by n-fa,c,6respectively. Hence show that eitherequationisintegrable infinite terms ifn=2<zorn2(n4-a).Byarepetitionofthisreasoning show that the firstequationofEx.9isintegrableinfinite terms if n2(sn +a),where(asalso inthefollowing examples)siszeroorany positive integer. xn (11)Show that thesubstitution 2=transforms theequationof Ex.9intooneofsimilar formwith a,6,creplaced byn-a,c,6respec- tively. Deduce that either isintegrableinfinite terms ifn=2(sn-a). (12)From theresults ofExs. 9,10,and 11deduce that Riccati's equationisintegrableinfinite terms ifm+2=2s(w4 2)2. Show that this result isequivalenttow=-4r/(2rl),wherer, like5,iszero orapositive integer,orto2/(rw+2)=*an oddinteger (positiveornegative). (13)Show thatthesubstitutionsy*=r-+~2v ^a?m^3 ,transform uXXJL TOTAL DIFFERENTIAL EQUATIONS 205 Riccati'sequationintoanother ofsimilar form with btc,mreplaced byc/(m-f3), 6/(m+3),-(m4-4)/(m+3)respectively. Deduce that if misoftheform-4s/(2s-l),thetransformationreplacessby(s-1). Byconsideringssuch transformations show that inthiscase Riccati's equationisintegrableinfinite terms. (14)Show that the substitutions y=l/Y, X~xm+l ,transform Riccati'sequationintoanother ofsimilar form with6,c,mreplaced by c/(m+1),6/(m-f1),-ml(m+1)respectively. Deduce(usingtheresult ofEx.13)that Riccati'sequationisintegrableinfinite terms ifmisof theform-4s/(2s +l). 168.Two methods ofintegrating the total differential equation Pdx4-Qdy+Rdz =Q.Wehavealready (inChap. XI.) giventhe necessary and sufficient condition ofintcgrabilityofthisequation, andageneral method ofobtainingtheintegral when thecondition issatisfied. Weshallnowgivetwoadditional methods. One of these(involvinganintegrating factor) hasthedefect that itcanbe usedonlyforcertain homogeneous equations, butfortheseequations itisperhapsthesimplest method available. Theother(Mayer's method)isquite general.Itrequires onlyoneintegration, andthis givesitatheoretical advantageovertheothergeneral method (Art. 117),whichrequirestwo. However, thebeginnerisnotadvised to usethismethod, forthesingle integration requiredisoften more difficult toeffect (onaccount ofthelack ofsymmetryoftheexpres- sions involved) than thetwointegrations requiredinArt. 117. Moreover, Mayer's method,ifappliedwithout careful attention to certain conditions, maygiveresults thatareabsolutely wrong. 169. Integrating factor forhomogeneous equations. Let Pdx+Qdy+Rdz=Q (1) beanintegrable equationinwhich P,Q,Rarehomogeneousfunc- tions ofthesamedegree ninx,y,z,that istosay,inwhich P,Q,B maybeexpressedintheforms xn f(u, v),xn g(u, v),xnh(u, v) respectively, where u~ylx,andWz/x. Then dy=udx+xdu, dz=vdx+xdv. Henceequation (1)becomes xn{f(u, v)dx+g(u, v)(udx+xdu)+h(u, v)(vdx-f-xdv)}=Q, i.e. %n{(f+ug +vh)dx +x(gdu+hdv)}=0, from which, dividing byxn+1(f+ug+vh),ifthisexpressionisnot zero,weobtain dxgdu+hdv_^ v+ ~ P.D.B. 206 DIFFERENTIAL EQUATIONS Now since equation (1)isintegrable,soisequation (2),either immediatelyoraftermultiplication byanintegratingfactor. But the firstterm inequation (2)involves only z,andthesecond term onlythevariables uand v.Onevariable isseparatedfrom theother two,and thisseparation,which isthemost favourable form for integration,would bedestroyed bymultiplication byanyfactor (exceptamere constant). Hence nointegratingfactor(excepta constant) canexist, soequation (2)mustbeexact asitstands. But, apartfrom thechangeofvariables, equation (2)wasderived from equation (1)bydivision bythefactor xn+1(f+ng+vh))which is equaltoPx+Qy +Rz. Hence I/(Px+Qy+Rz)isanintegratingfactor oftheintegrable homogeneous equation Pdx+Qdy+Rdz=Q, except whenPx+Qy+RzQ. Asimilar theorem holds goodfor theequationP1dxl+P2dx2+...+jPn<te==0. Ex.(y2+yz)dx+(zx+zz)dy+(t/2-xy)dz=0. Here Px+Qy+Rzxy2+xyz+xyz+yz2+yzz-xyz =y(%y+xz+z2+yz)=y(x+z)(y+z), sothe'integratingfactor isl/{y(x+z)(y+z)}. Multiplyingthedifferential equation byitweobtain dx zdy J^J7.?)*L x+zy(y+z)(x+z)(y+z) dx !x+zy(y+z) (x+z)(y~+z) dx dudy dz dz fir __ i__y__ y_j_______ AVJLi l "V/x+z yy+zx+zy+z dx+dzdydy+dz _ or -f--y--=0,x+z yy+z whence_log(x+z)+logy-log(y+z)=logc, giving y(x+z)=c(y+z). Examples forsolution. Applythismethod tothefollowing examples,2onp.138;10(i), 10(ii),and 11onp.144. 170.Mayer's method. Write thetotaldifferentiaLequationinthe form dz=P(x, y,t)dx+Q(x, y,z)dy. Itmaybeprovedthat ifthecondition ofintegrability (Arts. 118and TOTAL DIFFERENTIAL EQUATIONS 207 119)issatisfied, andifthefunctions PandQareholomorphicinthe neighbourhood ofapoint (^o^o^o)*then there exists onesolution (and only one)ofthedifferentialequation representingasurfacepassing throughthispoint.* Mayer's method determines thissurface by findingthecurve ofintersection ofthesurface andavariableplane drawnparalleltotheaxis ofzthroughthepoint (xyz).The simplestvalues consistent withtheholomorphiccondition aretaken forXQand?/ ;e.g.and 0,orand1,or1and 1.zoccurs inthe final result asthearbitraryconstant. Theprocedurewillbebest understood byastudyofthefollowing examples. (Ofcourse these examplescanbesolved atsight, but ifharder oneshadbeenchosen theprincipleofthemethod might havebeenobscured bythedetails of thecomplicated integrations which Mayer's method ofteninvolves). Ex.(i).dz=2xdx+ydy..............................(1) Thecondition ofintegrabilityis 2z(0-0)+4y(0-0)-1(0-0)=0, which issatisfied. Wemaytakex=0andyQ0,asthefunctions 2x and4t/areholomorphicintheneighbourhoodof(0,0,z).Theplane throughthispoint paralleltotheaxis ofzisgivenby y=mx,dy=mdx ..............................(2) Fromequations (1)and(2), whence weget Z-ZQ(\+2m2 )x2 ,..............................(3) determiningtheconstant ofintegration bythecondition that ZZQ whenz=0. Equation (3)representsacylinder (with generators paralleltothe axis ofy)throughthecurve ofintersection oftheplane (2)andthe surfacerequired. Eliminating mfromequations (2)and(3)wegetastheequationof thesurface This isthegeneralsolution ofequation (1),ifzistaken tobean arbitrary constant. n /.-v *3zdx 2zdy /A. Ex.(u). dz----^............................... (4) Thecondition ofintegrabilityis y y which issatisfied. Wecannot take z=*0,yQ^Q,asthismakes the functions3z/zand2zjyinfinite. However,ar=l,y=lwilldo. Goursat, Cour d?Analyse MatUmaiiqut, VoL II.,4thed.,Arts. 385and441. 208 DIFFERENTIAL EQUATIONS Put yl+m(-l)............................... (5) Equation (4)becomes 3zdx 2zmdx dz giving logz-logz=3logx-2log(1+m(x-1)}, whence z{l+w(x-l)}a=z3............................... (6) Eliminating mfrom(5)and(6),wegetthesolution zy*=*<??. Itwillbeobserved that allthesurfaces ofthisfamily passthrough thepoint (0,0,ZQ). Examples forsolution. (1)Show that theattempttosolve Ex.(ii)above, with(0.0,2) asthefixedpoint,breaks down when wetrytomake thecylinder correspondingtoequation (6)passthroughthatpoint. (2)Solve y*dz=ydx+(y*-x}dy. [Thecorrect result, choosingthefixedpointas(0,1,z)is y(*~2o)=3y(-1)+aj- Thechoice of(0,0,e)leads totheincorrect resultz-z=y.] (3)Solve(1+xy)dz=(1+yz)dx+x(z-x)dy. [Result2=05+Z(l4-?/).] 171. Linear differential equations ofthesecond order. The followingdiscussion(Arts. 171-177)issupplementarytoChaps. IX. andX. Suffixes willbeused todenote differentiations withrespect tox.Weshall useh(x), k(z),j(x), H(x) yK(x),orsometimes h,k,j, H,K,todenote functions ofxwhich areJiolomorphicattheorigin (i.e.expansibleinpowerseriesconvergentwithin asufficientlysmall circlewhose centre istheorigin)andwhich have thefurtherproperty thattheydonotvanish attheorigin.Theirreciprocalsalso willbe holomorphic,*andsowill theirlogarithmicderivates such as h^hix). Whenever wespeakofsingular points,itistobeunderstood that thesepointsareisolated,i.e.thatacircle ofsufficientlysmall radius withanyonepointascentre willexclude alltheothers. 172.Regular integrals. Itwasmentioned onp.110that solu- tions ofFrobenius' forms arecalledregular integrals. Weshallnow consider inmore detailwhat isimplied bythis. Letusexamine th forms oftheanswers totheexamplesinChap.IX.Although we *BromuictfsInfinite Series, 2nd ed.,Arts. 54and 84. LINEAR DIFFERENTIAL EQUATIONS 209 distinguishedfour*cases intheprocessofsolution, there wereonly twoessentiallydifferent forms ofthecomplete primitive au+bv. Oneintegral, say u,wasalwaysoftheform xa h(x). Thesecond integral, v,had insome examplesasimilar form, sayxft k(x),asin Arts. 95and99;inothers, asinArts. 97and98,ithadtheform xa {h(x) logx+x'k(x)}, where swasaninteger, positiveornegative (e.g.1inEx. 1,Art. 97, and-4inEx.1,Art.98). Wetake these forms asthedefinitions ofintegrals regularatthe origin f(ofalinear differentialequationofthesecondorder), with theslightmodification that sisallowed totake alsothevalue zero. Thismakes noreal difference, for ifsiszerowecanreplacethe integral v=x*{h(x) logx-fk(x)}bythelinear combination ofintegrals ,,X1 7/ x -=*ah(x)logx+%)" which isofsimilar formexceptthatk(x)hasbeenreplaced byanew holoniorphicfunction ofwhich xisafactor.Similarlyinthe first form ofv,namely x^k(x) ywecanalways suppose aand/3unequal, for ifnotvcanbereplaced byv-^u,which hasza4*asafactor. For linear differentialequationsofthemthorder anintegral regularattheoriginisdefined asoneoftheform xa {h(x)(\og x)r-fx*k(x)(log x)'*1+...+xn j(x)}, wheres,...narezero oranyintegers (positiveornegative), and r canhaveanyofthevalues 0,1,2,...w 1.Thus forfirst-order equations regular integralscannot involvelogx.Forthesecond *Inthemethod ofFrobenius forequationsofthemthorder(Crelle, Vol.LXXVI. 1873, pp.214-224, orForayth'sTheory ofDifferential Equation*, Vol.IV.pp.78-93, orluce's Ordinary Differential Equations, pp.396-402), itisconvenient forthe theoretical treatment todistinguish onlytwocases, thesecond ofwhich includes our cases II.,III.andIV.Todealwith thissecond casetheseries with itscoefficients asfunctions of*ismultiplied byf(c+1)/(c-f2).../(c+r),where /(c)=0isthe indicia!equation, and risthegreatestdifference between anytwo ofitsroots thatbelongtoasetdiffering byintegers (cf.ourmethod forcaseIII.). Inthis series and itssuccessivepartialdifferential coefficients with respecttocare substitutedrespectivelytheroots, arranged sothatthedifference between anyone andthefollowingisapositive integer orzero. However, insolving examplesthis method often leads toalargeamount ofunnecessary work, andhence inChap. IX. wehave modified itconsiderably, particularlyinourCase IV. tPoints other than theoriginareconsidered inArt. 175. Itisunfortunate that theword regular hasinDifferential Equationsameaningdifferent from that usual inTheoryofFunctions, where itisequivalenttoholoniorphic (asdefined inArt. 171). Thusanexpression involving logxorxa(whereaisnotzeroorapositive integer) mayboanintegral regularattheorigin, andyetcannot beafunction regularat thatpoint. 210 DIFFERENTIAL EQUATIONS order thelogarithmoccurs eitherlinearlyornotatall. Thismay alsobededuced fromChap.X.asfollows :InArt.107bothintegrals were freefromlogarithms.InArt. 110weobtained asecond integral bydifferentiating partiallywithrespecttocaseries of theformxcanxn ,where thea'swere functions ofc,andthen, after differentiation, replacingcby/?.Theresult (notgiveninArt.110)is which isoftheform x?{h(x) log(x)+x'k(x)} Ifthe firstXofthecoefficients an(/3)arezeroandalsothefirstJJLof c)flio\ thecoefficients -^-^,thena=/3-fX ands=/z-X.up Itwillbenoticed thattheco-factor oflogxisitselfanintegral. Thismaybeproved independently.Take thedifferential equation as yt+0iP(s)+!#(*)=o,........................... (i) where P(x)andQ(x)areuniform* (i.e.single-valued)intheneigh- bourhood oftheorigin. Ifintheleft-hand side ofthisequationwesubstitute forythe integralxa{h(x)logx+x*k(x)}=u logz-fw say,theresult must, by definition ofanintegral,beidenticallyzero. Inthis result logx occurs withaco-factor (u2+U-J? +uQ).Thisand alltheother terms intheresult, except logx,aretheproductofxaandauniform function, sinceuandwyandhence alsoul9u2,w },w2,areproducts ofthiskind, whilePandQareuniform. Ifwecould divide the identity bytheco-factor oflogx,weshould obtain theabsurd result- thatthenon-uniform functionlogxisthequotientoftwouniform functions,i.e. isitself auniform function. Hence thedivision is illegitimate,and thiscanbedueonlytotheco-factor beingzero; i.e.uisitselfanintegral. Asimilar theorem holds fortheco-factor ofthehighest powerof logxoccurringinaregular integralofanequation (withcoefficients uniform intheneighbourhoodoftheorigin)ofthemthorder. Thus ineverycase inwhich there areregular integralsatleastoneofthem must befreefromlogarithmsandoftheform xh(x). *This differential equation includes asparticular cases those considered inChape, IX.andX. LINEAR DIFFERENTIAL EQUATIONS 211 173.Fuchs* theorem. Thenecessary andsufficientcondition thatalineardifferential equation ofthesecond order, whosecoefficients areuniformintheneighbourhood oftheorigin ,should 'have allits integrals regularattheoriginisthattheequationshould beexpressible intheform wherepandqareholomorphicattheorigin. Thediscussion ofthemethod ofFrobenius(Arts. 106-110) proves that thiscondition issufficient. Wehavenow toprovethat itis necessary. From Art. 172, atleast oneintegralisoftheform xah(x). Denote thisbyu(x). Putyu\zdx, and substitute in equation (1)ofArt.172. Thetermsinvolvingthesignofintegration have afactor(w2+u^P+uQ)andtherefore vanish, asuisanintegral, andweget 2ulz+uz1+Puz~Q ...............................(2) Now theintegral ymayhave either ofthetwoforms (x),xa{h(x) logx+x'k(x)}. Hence =a^-,Or, u(x) h(x)& h(x) orlogx+x'H(x), say, sothat z=^)=*''{(/3- or x-1+xa~l(sH Inboth caseswecanwrite zintheform*x^K(x),where K(x)is holomorphicwith7i(0)^0. Hence fromequation (2) p=_?i_2w 1==_y_^i_2a_2A i=p(5) zu xKxh x*' wherepisholomorphicattheorigin. Also, sincexa h(x)isanintegralofequation (1), xah2+2aar~lhl+a(a- giving whereqisholomorphicattheorigin. Onmultiplyingeach sideofequation (1)byx2 ,andreplacing xP andx2Qbypandqrespectively, wegettheformrequired bythe theorem. *Inthe firstcase7=/3-a-l. Inthesecond case7=-!or-l, according astheintegersispositive ornegative. 212 DIFFERENTIAL EQUATIONS Example forsolution. Byeliminating thearbitrary constants fromy=Ax^ 4-Ex*log*, obtain thedifferentialequation 8x2 (4-logx)y2+2z(8-logx)y-ylogx-0, which istherefore alinear differential equationofthesecond order havingallitsintegrals regularattheorigin,but isnotexpressibleinthe formgiveninFuchs' theorem. [This example shows theimportanceofthestipulationthat the coefficients ofthedifferential equationshould beuniformintheneigh- bourhood oftheorigin.Infact, thisimposesasevere restriction, for itexcludes allcomplete primitivesoftheform y=AxPj(x) +Bxa {h(x) logx+x'k(x)}> exceptforthespecialcasewherex^j(x)ismerelyanumericalmultiple ofxh(x).] 174.Ordinary andsingular points. Itmayhappenthat (unlike theotherholomorphicfunctions A,&,j,H,K)pandqmayvanish attheorigin.Inparticularifpisdivisible byxandqbya;2 ,the equationinitsoriginalform(1)hasPandQholomorphicattheorigin. Inthiscasetheoriginissaidtobeanordinary point, andonapplying themethod ofFrobenius weshall obtain anindicialequation with and 1asroots, leading (asinArt.99)toanindeterminate coefficient andfinallytotwolinearly independent integralsthat are bothpowerseries. Neitherlogarithmsnorindices other thanpositive integers (orzero) canoccur. Buttheindicialequation mayhave and 1forroots without theorigin being anordinary point,asin Ex.2ofArt. 98. Points which arenotordinaryarecalledsingular.Ifata singular point (inwhose neighbourhoodthe coefficients ofthe equationareuniform)alltheintegralsareregular,itiscalled a regular singular point. These definitions refer tosingular pointsofthe differential equation itself, that is,ofitscoefficients when itiswritten inthe form(1).Ourdiscussion ofordinary points shows thatthesingu- larities oftheintegralsaresingularitiesoftheequation, butthe converse isnotalwaystrue. Forexample, byeliminatingthe arbitraryconstants AandBfromy=Axm+Bxn ,weget Ifmandnareunequal positive integers,orifone iszeroandthe other apositive integerother than1,theoriginisasingularityof theequationbutnotoftheintegrals. When, ashere, every integral LINEAR DIFFERENTIAL EQUATIONS 213 isholomorphicatapointwhich issingularfortheequation,the singularityissaidtobeapparent.Inallother cases thesingularity issaidtobereal. Atanapparent singularityitisnecessarythatthe roots oftheindicialequationshould beunequal positive integers,or zeroandapositive integer greaterthan 1.Itisalsonecessarythat thesmaller rootshould lead toanindeterminate coefficient(very much asinArt.99). Examples forsolution. (1)Show that anecessary (butnot sufficient) condition forthe origintobeanapparent singularityoftheequation wherep(x)andq(x)areholomorphicattheorigin,isp(0)anegative integer,while thenecessary andsufficient conditions fortheorigintobe anordinary pointarep(0) q(0) qi(Q)=0. (2)Show thattheoriginisanapparent singularityof andobtain thecomplete primitive (3)Show thattheoriginisarealsingularityof2 */2-i-(x2-2)y=0, butthat alltheintegralsarefreefromlogarithms. [Theroots oftheindicial equationare-1and 2.Thesmaller root givesa3indeterminate(cf.Art. 99;.The infinite series obtained canbe summed, giving finally yAx^^oa x+xsinx)+Bx~1(smx-x cosx).] 175.Equations ofFuchsian type. Todeal withpointsother than theorigin wemake achangeofvariable, puttingX=x-a, or X=x~1 ,accordingasthepointtobeconsidered isthe finite one x=a,orthat atinfinity x=oo .Itfollows that forequation (1),if thefunctions PandQareholomorphicateveryfinitepoint except alimited number a,6,c,...,then these aretheonly possiblefinite singular points.Thuswecanfindthesepoints byinspection, by seeingwherePandQfailtobeholomorphic,without makinga changeofvariable;e.g.if "D_. nr\(i f\ x(x-3)v~x2(x-3)(x-4)3' theonly possiblefinitesingular pointsaregivenbyx=0,3,4.More- over, totestwhether asingular pointx=a isregular, wehaveonly tonotice whether (x-a)P and(x-a)2Qareboth holomorphicat x=a. Intheexample given and3areregular singular points, but4isirregular,since(x~4)2 ()isnotholomorphicatx=4,owing tothefactor (x-4)inthedenominator. 214 DIFFERENTIAL EQUATIONS Thepoint atinfinity x=oo isbest dealt withbyachangeof variable. Ifallthesingular pointsofanequation (whosecoefficients are everywhere uniform)areregular,theequationissaid tobeof Fuchsiantype. Examples forsolution. (1)Show that, fortheHypergeometric equation theonly singular pointsare0,1andoo,which areregular. (2)Show that forLegendre's equation theonly singular pointsare 1,-1,and oo,which areregular. (3)Show that forBessel'sequation theonly singular pointsareand oo,ofwhich the first isregular,but notthesecond. fabc } (4)Show thatRiemann'sP-equation y=P\afiyx\t (a'/3V J 1"a"a x_a x_a X-a)X-)(x-o has a,6,casregular singular points and allotherpoints, includingoo, asordinary points, providedthata-f-a'-t-/3 +/3'+y+y'=l. Bychangeofvariable show thataandaaretheroots oftheindicial equation correspondingtothepointa. (5)Show thattheequationsofExs. 1,2and 4,butnot 3,areof Fuchsiantype. (6)Show thatthefollowing equationisofFuchsian type: where\fsistheproductofanynumber, say n,linear factors (x-a) t (x-b), (x-c),...ofwhich notwoareequal, andP,Qarepolynomials inxofdegrees notgreater than (n-1)and(2n-2)respectively. 176. Characteristic index. Consider theequation y2+x~x p(x)ij 1-fz-^(%=0, where X,^arepositive integersorzero,andp,qareholomorphic functions ofxwhich arenotzerowhen x=0. Ifweattempttosolve thisequation bythemethod ofFrobenius, wegettheindicialequation byreplacing ybyaseries ofpowersofx (startingwithx*),andequatingtozerothecoefficient ofthelowest LINEAR DIFFERENTIAL EQUATIONS 215 powerofxintheresult furnished bytheleft-hand sideofthediffer- entialequation. Thelowest powersofxfrom itsfirst, second, and third terms willberespectively c-2,c~A-l, andc-^u. Three cases arise : (i)ifthe first ofthesenumbers isnotgreaterthan either ofthe others, theindicial equationisofthesecond degree ; (ii)ifthesecond ofthesenumbers islessthan the firstandnot greaterthan thethird, theindicialequationisofthefirst degree. (Cf.Exs.2and4,p.118) ; (iii)ifthethird ofthese numbers istheleast, then theindicial equationisofzerodegree. (Cf.theexample atthetop ofp.118). Incase(i)X<1and/x<2,sobyFuchs' theorem there mustbe tworegular integrals. Incase(ii)theremaybeoneregular integral. If,however, asis often thecase(cf.Ex.4,p.118), thesingleseries obtained isdivergent forallvalues ofx,there isnoregular integral. Incase(iii)there isnoseries andhence noregular integral. The characteristic indexmaybedefined asthenumberdenoting thecasewhich arises, butstarting from zero, i.e. forcase(i),1for case(ii),and2forcase(iii).Itiseasytoextend thisdefinition and thediscussion ofthemaximumpossible degreeofthe indicial equationtoequationsofanyorder, leadingtotheconclusion that alineardifferential equation ofordermand characteristic index r cannot havemore thanm-rregular integrals. 177.Normal andsubnormal integrals. WesawinArt.100that themethod ofFrobenius failed todiscover anintegral withafactor i e*.This isaparticularcase ofanormalintegral,defined asoneof theform e*u,where zisapolynomialin1/x(inthesimplestcasea numericalmultipleofl/#),anduisafunction ofxsuch asoccurs in aregular integral. Subnormalintegralsdiffer fromnormalintegrals onlybyhavingxreplaced byitssquareroot(orbyitscube orother higher root inthecase ofdifferentialequationsoforderhigher than thesecond). Amethod ofobtainingnormal orsubnormalintegralsisshown bythefollowing examples: Ex.(i)., t/2~2ar1y1+ar*(~4 +2a;%0 (1) Here theindicialequationhasnoroots andthere arenoregular integrals (i.e.thecharacteristic index is2).This isduetotheterm -4ar4inthecoefficient ofy. 216 DIFFERENTIAL EQUATIONS Put y=e*u, giving yl-e8^4-Zjtt), t/2-e*{u 24-2*^-f(z Equation (1)istransformed, after division bye*,into Togetridoftheterm-4ar4 ,take zlasaar2 ,wherea= 2.Equa- tion(2)becomes u24-(-2ar14-2aor2)w14-(2x~2-4aar3)u=0, which hasacharacteristic index 1,andsomayhave aregular integral. Applyingthemethod ofFrobenius tofind this,wegetthesimpleresult u=x2forboth values ofa.Multiplying bytheexponential factor, we obtainfinallythetwonormalintegralsx2e~2lxandxze2/x . Ex.(ii),y24-4ar2y14-ar6 (-44-6x2-4a%-0. Againthere arenoregular integrals. ProceedingasinEx.(i),we obtain u24-(4ar24-2z1)w1-i(-*ar64-6or4-4ar84-4x~2z14-Zj24-z2)u0. Togetridoftheterm 4X"8 ,take zltocontain aterm for3 ,where 6=^2.Ifzl=ax~24-6jr3 ,thecoefficient ofuwillcontain noterm in x""5 ,providedthataischosen sothat4&4-2a& =0,i.e.a=-2. Thechoice zx=-2ar24-2ar3leads to which hasoneregular integral, u=x. Theother choice, z1=-2ar2-2x~3 ,leads to u2-4x~3w14-8x~%1==0. Thishasnoregular integral,fortheonlyseries obtainable, namely, .,.1.3.1.3.5 . Isdivergent. Hence theoriginal equationhasonenormalintegral, Ex.(Hi). y2+ar2 (-1+3x)y i4-ar2y=0. Thistimethecharacteristic index is1.Theindicialequationisofthe firstdegree, but(aspointedoutinEx. 4,p.118)theseries obtained is divergent. Proceedingasbefore, weget u2+(-or14-32T14-201)u14-{or24-(-or24-Sar1 )*!4-Zj24-z2}w=0. Asthetroublesome term intheoriginal equation was-x~2inthe coefficient ofyvwhile thecoefficient ofywasonlysuch asoccurs when theintegralsareregular,itmightbethoughtdesirable tosimplifythe coefficient ofubytakingzl=Jar2 .Butthiswillintroduce aterm inaH intothecoefficient ofw,giving anequationwithnoregular integrals. Letustrytogetanotherequation with characteristic index 1,in thehopethat thecorrespondingseriesmay converge. Put2^=oar2 . LINEAR DIFFERENTIAL EQUATIONS 217 The coefficient ofuwillbefreefromterms inor4ifa2-a 0,i.e.a~0 or1.a=0givestheoriginal equation,buta=lgives u2+(32T1+x-2)^+(x-2+x-3 )u-0, which hastheregular integral u**x~l ,givingtheonenormalintegral Ex.(iv). ya+lor1^-x~*y=0. Thisequationhasnoregular integrals. Proceedingasbefore, weget uz+(lx~l+%*i)ui+(-xr3+Jx-^i4-Z!2+z2)u=0. Togetridoftheterm-or3 ,take zl=&x~3/2 ,where A-1. Thisgives 2+(Jar1+2fci;-3/2)ui-*"5/2w-0. 00uxc^janxinwillbeanintegralif o ac(2&c-&)=0, sothat c= , ai{2k(c+J)- A;}+a{c(c-1)-f\c]-0, i.e.^-1-0=0,soa^O. Similarly, an=forallvalues ofn>l, sow=x*. Theoriginal equationhasthetwosubnormalintegrals and Examples forsolution. Find normal orsubnormalintegralsofthefollowing equations (1M5): (1) (2) [^4n5.xW*9xh~ilx ;orx*cos(1/x),x*sin(1/x).] (3)y2-fx~2 (-2+x)y!+x~4 (l-fx-x2+x4 )y=0. [/Ins.we~1/4; ,ve~1/a; ,where uandvareasonp.115.] (4)y2-ix~1 i/1-4x-3y=0. [Ana. x(l+|xtyr4*-*,x(l-Jx*)e4x "*.] (5) t/2-x-6(l+5x%=0. [ylns. x~1 (l+Jx2)^*^* ;z=~|x~2givesadivergent series.] (6)Transform Bessel'sequationoforder zerobythesubstitution xaasl/JSC, andattempttofindnormalintegralsofthetransformed equation. Show thattheseries obtained aredivergent. Revertingto theoriginal variable, obtain theseries 1Six 2!(Six)23!(Six)3' andasimilar series with thesignofichanged. [The transformed form ofBessel'sequationisgivenintheanswer toEx.1,p.118. These series, although divergent,areveryuseful. They arecalled asymptotic. Foranygiven value ofx,sufficiently large, they givean 218 DIFFERENTIAL EQUATIONS approximation whose error canbemade reasonably small, thoughnot indefinitelyso.SeeWhittaker andWatson's ModernAnalysis, 4thed., Arts. 8-1-8-32 and17-5.] (7)From Whittaker's confluent hypergeometric equation k- / .-m y.+(-i+i+V~ obtain (bytheprocessofEx.6),theseries e-t^Fl +yK-^-OT{^-(t- L/=!r\xr\xr [Thisseries isingeneraltheasymptotic expansionofthefunction denoted byWt>m(x),but if(k-$m)isapositive integertheseries terminates, giving anintegralinfinite terms. Another seriesW_j_m(-x) canbeobtained fromW^m (x)bychangingthesignsofkand x.] 1/8.Theequation ofvibrating strings. This is 132F cte2a23<2' where aisaconstant. PutX=x-at, T=x+at. Th ^Z-^ZM ^?^_^r BCB~3Z 3x+VT3~ ' VV_-d/3F\_/_3.^-^V^/-\3Z 32F o- -i3F_9F3Z3 bimilarly--.-+.(1) Substitutinginequation (1),weget giving2j, andV=f(X)+^<j>(T)dT, or t.c. V=f(x-at)+F(x+at),........................ (2) where/andFarearbitraryfunctions. MATHEMATICAL PHYSICS 219 f(x-at)isunaltered if#isincreased byaand tby1;hence it represents awavemoving alongthepositivedirection oftheaxis ofxwithspeeda.Similarly F(x+at) representsawave moving alongthesame linewith thesamespeedintheoppositedirection. Analternative method ofsolving equation (1)istousethegeneral result giveninArt. 145,with x,y,zreplaced by t,x,Vrespectively. Writingtheequationas or (D2~a2D'2)F=0, wegettheauxiliary equation m2-a2=0,whose roots are-aanda, leadingto V=f(x-at)+F(x+at). 179. Particular solutions oftheWave equation. This is 327327 B2F !327 ' where aisaconstant. Itisthethree-dimensional analogueofthe one-dimensionalequation (1).Letusattempttofindasolution similar to(2),butwith x,y,z,tinstead ofx,t. TryVf(h -{-my+nz-at)+F(lx -{-my+nz+at),...........(4) whereI,m,nareconstants. Equation (3)issatisfied if InthiscaseI,m,naretheactual direction-cosines ofacertain line. The firstfunction isunaltered ifx,y,z,tareincreased byla,ma,na, 1respectively,soitrepresentsaplane wave (whose normal has direction-cosines lym,n)moving paralleltoitself withspeeda. Thesecond functionrepresentsaparallel wave movingwith the samespeedintheoppositedirection. Henceequation (4)represents thepropagationofplanewaves. This isoneparticularsolution of theWaveequation. Toobtain asolution forspherical waves transform equation (3) intospherical polarcoordinates. Thework isessentially atrans- formation ofLaplace's equation,*andweget *SeeEdwards'Differential Calculus, Art. 532, or,forasimpler methodusing Gauss1theorem, anybookonAnalyticalStatics. 220 DIFFERENTIAL EQUATIONS Forasolutionsymmetricalinalldirections about theorigin, i.e.independentofand<,thisreduces to Bythetransformation V=rV, weget 9r' soequation (6)becomes, aftermultiplication byr, &U_l'&U 3r2~a23*2' giving U-f(r-ot)+F(r -fat), (7) Thisrepresents twospherical waves with thesamespeed a,one divergingfrom theoriginandtheother approachingit.Thefactor 1/rshows that theintensityofthedisturbance decreases asthe distance from theoriginincreases. 180. Poisson's (orLiouville's) general solution. This obtains V atanytime tatapointPinterms ofthemean values overasphere ofcentrePandvariable radius atofthefunctions, saygandG, 3Fwhichgivethevalues ofVandrespectively when J=atany pointinspace. Takespherical polarcoordinates withPasorigin. Now themean value/ofafunction/(r, 0,0,t)overasphereof radius risgiven by Take themean value overasphereofradius rofeachterm ofthe Wave equation (5).Thesecond termbecomes rr2 JoJo;oJo'Ul/ x uiy/ *'"JoL^Jo andthethird MATHEMATICAL PHYSICS 221 Both arezero, forsin6vanishes atboth limits, while </>=2?rgives the same value of as0=0 (whichisreallythesameposition). The firstandfourth terms donotvanish. These give 13/.3F\ 1-&V.^tt\r --sfr'a i?'..........................(8) BOthat rV=f(r-at)+F(r+at),..........................(9) =/(- <ti)+lf(aO+r{f'(-at)+F(at)} +W'(~at) +!'(<*)} +.................................... (10) IfFistobefinite attheorigin (r=0)forallvalues oft f(-at)+F(at)=Q, Hence, fromequation (10), using asuffix todenote theresult ofputting r=0, F=/(-oO+F(a*)=2F(aO........................(11) Fromequation (9), and r^-- af'(r-at)+ aF'(r +at),Ot whence 2F'(r+at)=^- (rV)+--~ ,uT (Iut forallvalues ofrand t.Putting (=0,andusingthe initial con- ditions, weget whence, givingrthespecialvalue at,andusing equation (11), ButF,theaveragevalue ofFoverasphereofzero radius,is simplyF . Thus F!($)+ Itfollows from theform ofthissolution that atanytime, t,the value ofFatanypointPdepends onlyupontheinitial disturbance atpointsonthesurface ofasphereofcentrePandradius at.Inan F.D.B. a 222 DIFFERENTIAL EQUATIONS explosion the initial disturbance isgenerally confined toaregion bounded byaclosed surface 8. IfPisexternal tothissurface and Aistheshortest distance fromPtoS,noeffect willbeproducedatP until atimed/ahaselapsed,forbefore then thesphereconcerned willgoonlythrough regionswhere there isnoinitial disturbance. Atanytime ItheWave-front (thelocus ofpoints justreached by thedisturbance)isasurface obtained fromSbyproducingallthe outward normals adistance at. Othergeneralsolutions oftheWaveequationhave been given byKirchhoff*(whose form isofimportanceinOptics), Whittaker, f andBateman.J Example forsolution. Verify that F I If(xsinucosv+ysinusinv+zcos ti+at,ti,v)dufo, J-irJ -IT where thefunction /issuch that differentiations under thesignof integrationarelegitimate,isasolution oftheWaveequation. [This isWhittaker'ssolution.] 181.Other differential equations ofMathematical Physics. These includeLaplace's equation 3z2Vas2 Poisson'sequation 327 theequationoftheconduction ofheat theequationoftelegraphy Schrodinger's equation (ofWave Mechanics) P~' ofwhich, inaparticular case,asolution isindicated intheexample attheendofthis article. *SeeJeans, Electricity andMagnetism (6th ed.)Art. 580,orDrude, Theory of Optics (translated byMann andMillikan), p.179. Foraphysicaldiscussion of another equation connected withwavepropagation,seeJeans, Art. 645. tSeeWhittaker andWatson, Modern Analysis (4th ed.), Art. 18-6. JIbid.p.402. MATHEMATICAL PHYSICS 223 Theseequations may bediscussed fromtwopointsofview. Treatises onpuremathematics*givealogicaldiscussion ofthe general solutions, butthephysicist complainsofthegreat lengthof thediscussion, andofthedifficultyofapplyingthesegeneralsolu- tions. Ontheother hand, treatises onphysics useacombination oflogicandintuition toobtain solutions(usually particularrather thangeneral) which haveaphysical meaning, andmightnever have beenreached atallbylogicalone. There isusuallylittledoubt that these results aresubstantially correct, butanyuncertainty, howeverslight,isrepugnanttothe puremathematician.Probablyhisknowledgeoftheunreliability ofintuition inpuremathematicsprevents himfromappreciating thevaluable andgenerallyreliablepartthat ithasplayedinphysics. Eitherpointofviewrequiresaveryextensive treatment, which cannot begiven here.f [Themoreelementary equationsofmathematicalphysicshave been dealt with inseveralplacesinthisbook, e.g.pp.24,28,29,36, 46-48, 49-61, 189, 190,234,235,241-247, 250, 251.] Example forsolution. FromSchrodinger's equation,withA/2?r replaced byK,andVgiven thespecial form-e2 /r,obtain, bychanging from Cartesian tospherical polar coordinates, replacing \fsbyr-lU(r)S(9, 0), (cf.Art.179), BytakingrlStobeasolution ofLaplace's equation (andhence asolution ofwhatourlastequationbecomes whenmisreplaced byzero), obtain * dr* Finally, bythesubstitutions reduce ittoWhittaker's confluent hypergeometric equation (Ex. 7, following Art. 177),withUtEtand(I+)inplaceofytx,mrespectively. [Forthephysical meaningofthiswork seeBiggs, Wave Mechanics.] ' e.g.Goursat,Cours d*Analyst Mathematique, Vol. TIT. fSeeRiemann- Weber, Partielle DifferentialgleichunyenundderenAnwendung aufphysikalische Fragen (thelatest edition hasbeenquite transformed, andbears the title Die Differential-und IntegralgleichungenderMechanik undPhysik); Jeffreys, OperationalMethods inMathematical Physics (Heaviside's methods); Picard, Lemons surQuelquts Type* Simples #Equations auxDerivees Partidles avec desApplicationsalaPhysique Mathematique;Webster, Partial Differential Equa. tions ofMathematicalPhysics;Bateroan, Partial Differential Equations ofMalhe- maticalPhysic*, 224 DIFFERENTIAL EQUATIONS 182.Numerical approximation. Adams' method. Resumingthe subjectofChapterVIII. weshallnowgiveamethod*which Prof. Whittaker considers tobethebest ofallthose tested inthe EdinburghMathematicalLaboratory.Itmaybeshortlydescribed asthecombined useofTaylor'stheorem andofacertain formula, given below, belongingtotheCalculus ofFinite Differences.Taylor's series isused forincrements ofxsmall enoughtomake theseries converge rapidly.After thusobtainingafew(generally four) values ofywehave sufficient data toobtain further values from theDiffer- enceFormula, thusavoidingtheuseofTaylor'sseries forlarge increments ofx.Theerror inthefinal resultmaybeestimated bya methodexplainedbelow. Ex. Given thedifferentialequation x^--ft/~2x=0,with theinitial values x=2, t/=2-5,findthevalues ofycorrespondingtox=2*05, 2*10, 2-15, 2-20, 2-25, 2-30, 2-35,240, 2-45, 2-50,andestimate theorder of theerrors intheresults. Weshall usehtodenote theincrement ofx,xnfor(x-f-nA), andyn forthevalue ofycorrespondingtoxn. Thesuccessive differential coefficients ofywithrespecttoxwillbe denoted byy',y" ,y'",...andtheir initial values bythesuffix . Todetermine thecoefficients intheTaylor'sseries putsc=2,y-2'6intheoriginaldifferentialequation andintheresult* ofdifferentiatingitsuccessively. Weget -, -o andsoon,leading finally to y.2l+l(x-Z) +l(x-W-j,(s-W +M*-W-M*-*r +~'(1) Ifweputinsuccession x2-05, 2*10, 2-15, 2*20 inthis series, the numerical value ofthelastterm written there willbe,atitsgreatest, ^T(0-2)5=0-000005, sothecorrespondingvalues ofywillbecorrect tofiveplacesofdecimals. Thusweget ft-2-53780, y2-2-57619, ya-2-61512, y4-2-65455. *Due toJ.C.Adams anddescribed inTheories ofCapillary Action, byF. Bashfortb and J.C.Adams. SeealsoChap. XIV. ofTheCalculus ofObservations, byE.T.Whittaker andG.Robinson. JohnCouch Adams, ofCambridge (1819-1892)isbestknown byhisdeduction oftheexistence ofthethenunknown planet Neptune from theperturbationsol Uranua. NUMERICAL APPROXIMATION 225 WenowusetheDifference Formula* Sf+i-y-? +tAgB.1+AA1jn-iH-fAVi +H*Al s1l-4-f...(2) whereqndenotes thevalueoth-^-whenxxn,y~yn,soinourexamplectx A^denotesjfn+1-jfn> A2gndenotes Ajn+i-Agn,andsoon. Putting n=*5, equation (2)gives y5*sy4+J4+iAg,+AA^ 1+|A8 j1+*Al j+...............(3) NowqQ-0-05(2- t//z )-0-03750. Similarly jt=0-03810, q2=0-03866, q3=0-03918, y4=0-03967. Hence A^=^j-<7=0-00060, and soon.Forthecalculation ofthese differences itisconvenient towrite thenumbers intheform ofthe followingtable : 0-00001 Letusexamine thenumerical value ofthevarious orders ofdiffer- ences shown inthis table. OnpassingfromAgtoA2 grwefind a decided decrease. Butthere isonlyaslightfurther decrease inA3 , andnone atallinA4 g.ThissuggeststhatA3 <jandA4 gareinaccurate. Wetherefore disregard themandapply equation (3)intheapproximate form. -2-65455 +0-03967 +0-00025-0-00001 -2-69446, Theerrorduetotaking onlyfourterms oftheseriesmaybeexpected tobedistinctlylessthan thelastterm retained, andthereforenegligible tofiveplacesofdecimals. Ontheother hand, althoughthetruevalue of thefirstandsecond terms cannot difier from theirrespective five-figure *This isobtained byintegrating withrespecttor,between thelimits and1, theinterpolation formula SeeWhittaker andRobinson's Calculus ofObservation*, p.365. 226 DIFFERENTIAL EQUATIONS approximations bymore than0*000005, these errors may,inanunlucky case,bedoubled inAganddoubled againinA2 g.Even ifevery term us^d inthecalculation ofy5had itsgreatest possible error, and ifthese errors alloccurred withthesamesign, theresultingerror iny6would be lessthan 0-000025. Wenow calculateg6=0-05(2 -t/5/z5)=0-040 12.Thiscanberelied uponasaccurate tofiveplacesofdecimals, asanerror of0-000025 inya would bemultiplied bythesmall number0-05/2-25,and sobecome negligibletoourorder ofapproximation. Addingthevalueq$toour tablewecanatoncegetA<?4=0-00045, andA2^^-0-00004, andhence -2-69446+0-04012 +0-00022-0-00002-2-73478. (Asthelastdigitisodd forbothA^3andA<?4,inhalving wehave to choose between twoequally good five-figure approximations. We choose thelarger andsmalleralternately,soastoprevent anaccumula- tionoferrors.) Proceedinginthisway,weobtain theresultsgiveninthefollowing table: y 9 AJ Afj y=2-50000 g=0-03750 ^=2-63780 ^=0-03810 t/t=2-576195,=0-03866 y3=2-61612 q3=0-03918 y4=2-66455 q4=0-03967 y6=2-69446 q6=0-04012 ye=2-73478 ?6=0-04055 y7=2-77554 g7=0-04095 y%=2-81668 q%=0-04132 yg=2-85817 qg=0-04167 y10=2-90001 They'smaybeexpectedtohave small errors inthelastdigit. As amatter offact, thedifferential equationthatwehave chosen hasthe exact solution y-x+l/x. Calculating from thiswefindanerror of 0-00002 iny&0-00001 iny7,yQ,y9Jyw,andzero intheothers. Toobtaingreater accuracy wemaycalculate yvy2,y3,y4,tomore placesofdecimals, sayeight. Thestudent should dothis. Itwillbe NUMERICAL APPROXIMATION 227 found thatA#,A2 g,A8gandA4 gallappeartobereliable, andsocapable ofuseinthedifference formula. The final results are y=2-500,000,00; ft=2-537,804,88; yt=2-576,190,48; y3=2-615,116,28; y4=2-654,545,45; j/6=2-694,444,42 (error-2inlastdigit) ; j/6=2-734,782,58 (error-3inlastdigit) ; y7=2-775,531, 88(error-3inlastdigit) ; y8=2-816,666,61 (error-6inlastdigit) ; yg=2-858,163,23 (error-4inlastdigit) ; y10=2-899,999,93 (error-7inlastdigit). The lastterm used inthecalculation oft/10,namelyf^A4^,has thevalue -0-000,000,09. Themagnitudeofthis indicates that the errors thistime (unlike those forthefive-figure work) probablyoccur fromneglectofthehigherdifferences. Toremedy this,wecaneither calculatey5accurately from theTaylor's series, anduseA5 g,or(asis moreusual) diminish theintervalsufficientlytoensure thatA6 </may benegligibletoourdesired order ofapproximation. 183.Remes' extension ofthemethod ofArts. 90-93. E,Remes hasgiven*asystematic method ofdeterminingsuitable values foi thenumbers mandMdefined inArt. 92,namely, Case(i)m=/(a, 6),M=/{a+A,6+A/(a +A,6+A)},if dfldx>0, 3//cty>0; Case(ii)i=/(a, 6),M=/{a+A, b+kf(a, 6)},if df/dxX), 3//fy<0; Case(iii)m=f{a+h,.b+hf(a+h,6-A)},M=f(a, 6),if Case(iv)w=/{a+A, 6+/(a, 6)},Jtf=/(a, 6),if dfjdx<0 93//3y<0. These valuessatisfy theinequalities (7), (8), (9),(10)ofp.107. Remes shows that ifwedefineRandrbytherelations r=JA{/(a, 6)+f(a+A,6+m*)} fR=$h{f(a, b)+/(a+A,6+Mh)}, theinequalities hold alsowhenqisreplaced byrandQbyR. *Phil.Mag., Series 7,Vol. 6,Feb. 1928. 228 DIFFERENTIAL EQUATIONS Let2'denote(p+2Q)if >0, butl(P+2 ?)if|<g<0. Let2"denoteJ(2p+R)if butl(2P +r)if|g<0. ThenRemes provesthattheerrors intheapproximations2'and 2"areatleast ofthefourth andthird orderrespectively (takingthe increment tobesmall ofthe firstorder)if^-J-Va<0, butatleast 'oyaxax* ofthethird andfourth ordersrespectivelyif~-~\ ^"T2>^-This conclusiondepends uponmandMbeingchosen asexplainedabove. Theerror intheexampleonp.107wasmuch smaller thanwould beexpected from this result, butthisseems tobeduetoluck inthe choice ofmandM,which werenotobtained intheway stipulated byRemes. Ingeneralthemethods ofAdams orKutta seemmuch better. APPENDIX A Thenecessary and sufficient condition thattheequationMdx+Ndy**Q should beexact (a)Iftheequationisexact,Mdx-fNdy=aperfectdifferential =df,say. So M-% andN-%;ux dy ^ f dN a2/ a2/dM therefore^-=.^*-=5-^-*=-5-,axaxe?/ cty&c a?/ BOthecondition isnecessary. (b)Conversely,ify^^-*PutF=*\Mdx,where theintegration isperformed onthesuppositionthatyisconstant. -.M i^^jLnen ~^ M.anu ^^ ^^~ ^ ~^ox oxdyoyoxoyox OEl 2V_-=aconstant asfaras a;isconcerned, that is, ^afunction ofy, Then Nowput / j Then N=y.dy flJf AlsoM=-5-bydefinition ofF ox =^-,sinceFand/differ onlybyafunction ofy. ThusMdx+Ndy=dx-f yj-dy rf/,aperfectdifferential. Sotheequationisexact, that is,thecondition issufficient. d2fB2f [Ourassumptionthata~a^pprigjustifiedif/and itsfirstand secondpartialdifferential coefficients arecontinuous. SeeLatnb'j Infinitesimal Calculus, 2nd ed.,Art.210;or3rded.,Art.1S3.] 229 APPENDIX B Theequation P(x ty,z)J-+Q(x,y,z)-t+R(x,y,z)=0,regardedcw four-dimensional,hasnospecial integrals. (SeeArt. 127.) Let u(x, y,z)=a, v(x,y,z)=b, beanytwoindependent integralsoftheequations dx/P Thenweeasily provethat and P+Q+R=.-..............................(2)dx By dz Theleft-hand side of(1)doesnotcontain a,andtherefore cannot vanish merelyinconsequenceoftherelation w=a.Hence itmust vanishidentically. Similarly equation (2)issatisfiedidentically. Naw letf=w(x,y,z) beany integraloftheoriginal partial differentialequation,sothat .o............................... (3)dx dy dz This isanother identicalequation,since/doesnotoccur init. Eliminating P,Q9Rfrom(1), (2), (3),weget ^4=0 identically.o(x 9y,z) Hencewisafunction ofuandv,say w= (f>(utv). Thati&jfwispartoftheGeneralIntegral, andtherefore, as/=t0 isanyintegral,there arenoSpecial Integrals. [The student willnotice theimportanceintheabove work ofa differentialequation beingsatisfiedidentically.Hill'snew classification oftheintegralsofLagrange'slinear equation (Proc. London Math. Soc. 1917) draws asharp distinction betweenintegralsthat satisfyan equation identically and^those which havenotthisproperty.] APPENDIX Theexpression obtainedfordzbyJacobi's methodofsolving asingh partial differential equation ofthefirstorder(Art. 140)isalways integrable. Toprove that dz=pldxl+p%dx 2+p^dx 3 isintegrableitisaccessary and sufficient toprovethat L=M=^-0, ...................................... (A) where L-dp~*-^M?-8-^N=^ -%WllL-lt? Ll-_ _y1X1.=- -IJT=A - OX3OX2 OX1OXZ OX2OXl Now, byadding equations (8), (9),(10)ofArt.140andusingthe relation (F,FJQ, butnotassumingthetruth of(A),weget (B) SimilarlyL'- +M1+2V =.............. (C) andL+M*+N*- ............. (D,9(^2>? 3) d(p*Pi) ^(P^Pz) From equations (B), (C),(D)weseethat eitherLMN=0 or A=0,whereAisthedeterminant whose constituents arethe coefficients ofL,MyNin(B), (C),(D). Butthese coefficients arethemselves theco-factors oftheconstituents ofthedeterminant andbythetheoryofdeterminants A=72 NowJcannot vanish,* forthiswouldimplytheexistence ofa functional relation which would contradict thehypothesisofArt.140 thatthep'scanbefound asfunctions ofthe a?'sfrom F^Ft-a^Fz-a^Q. Hence A=fO;therefore L=1/=N=0. *Alltheequationsofthisappendixaresatisfiedidentically. 231 APPENDIX D Suggestions forfurther reading -Noattemptwillbemade here togiveacompletelistofworks on differentialequations. We shall merely give thenames ofavery smallnumber ofthemostprominent, classified inthree sections. I.Chiefly ofanalytical interest(forming acontinuation toChapter X.). (a)Forsyth:TheoryofDifferential Equations (1890andlateryears, Cambridge Univ.Press). Thisimportant work isinsixvolumes, and isthemost exhaustive treatise inEnglish upon thesubject. Itshould notbeconfused with hismoreelementary work inonevolume (4th ed.1914, Macmillan). (b)Goursat :Cours d"Analyse mathtmatique,Vols. II.and III.(2nd ed.1911-15, Gauthier-Villars;Englishtranslationpublished byGinn). This deals almostentirely with existence theorems. (c)Schlesinger:Handbuch derTheorie derlinearenDifferential- gkichungen (1895-8, 3vols, Teubner). II.Partly analytical butalsoofgeometricalinterest. (a)Goursat :liquations auxdtriveespartielles dupremier ordre(1891). (b)Goursat :Equations aux deriveespartielles dusecond ordre (1896-98, 2vols.,Hermann etfils). (c)Page:Ordinary differential equations fromthestandpoint ofLie's Transformation Groups (1897, Macmillan). This deals with theelements ofdifferentialequationsinahighly original manner. III.Ofphysical interest(formingacontinuation toChapters III.andIV.). (a)Riemann :PartielleDifferentialgleichungen undderenAnwendung aufphysikalische Fragen (1869, Vieweg). (6)Riemann-Weber :Arevised edition of(a),with extensive additions (1900-01, Vieweg). (c)Bateman :Differential Equations (1918, Longmans). Thiscontains many references torecent researches. Itisimpossible tomentionoriginal papersinany detail, butthe recent series ofmemoirs byProf. M.J.M.HillintheProceedings ofthe London MathematicalSociety should notbeoverlooked. Addenda(publishedsince1920). I.(d)Ince :Ordinary Differential Equations (1927, Longmans). I.(e)Bieberbach :Differentialgleichungen (2nd ed.,1926, Springer). II.(d)Dickson :Differential Equations fromthegroup standpoint (1924, Princeton Univ.Press). Forother references seethesecond footnote toArt. 181. Thenew editions ofI.(b)andofForsyth's one-volume work areverylittle altered. 232 * MISCELLANEOUS EXAMPLES ONTHEWHOLE BOOK [London.]dy dx (2)dx (3)tany~~-I-tanxcos ycoa*x. (5)(l- (6) (7) (8) du (9)cosxsinx~y+cos . /Z (10)= +1. (13) (14)I (15) ( (16)1xydx (17)a3>~2 (18) (x+2y-*)p+(3y-z)}-x4-y, 233[London. ] [London.] [London.] [London.] [London.] [London.] [London.] [London.] [London.] [London.] [London.] [London.] [London.] [London.] [London.] [London.] ILondon.] 234 DIFFERENTIAL EQUATIONS (19)Wg+:-0. [London.] (20)p(x+p)+q(y +q)**z. [London.] (21)r+8**p. [London.] (22)z-%px-qy**p*/x*. [London.] (23)r-x=*t-y. [London.] (24)z=>px+qy-sxy. [London.] (25)z(rt~s2)+pqs=*Q. [London.] (26)x2r+2xy$+y2t=xy. [London.] (27)rq(q+l)-s(2pq+p +q+l)+tp(p +l)=*Q. [London.] (28)f=xy*p+x*p*. [Math. Trip.] (30)|-~ +x*ny~0. [Math. Trip.]ax xax (31) (zp+x)2+(zq+y)2~l. [Math. Trip.] (32)Find asolution oftheequation -j-|-3~-f2y=e3*which shaU vanish when x=andalsowhen z=log2. [Math. Trip.] (33)Solve theequation d2xndx,,.,.. -jp+2/c-T+(/c2+X2 )x=Acospfc Show that, fordifferent values of7),theamplitudeoftheparticular Integralisgreatest when7?2=A2-/c2 ,andprove that theparticular integralisthen (A/2K\)cos(pt-a),where tana=>p/K. [London.] (34)Solve theequation d2udu y-~:+tanaj-f ycos2z=*0dx2dxJ byputtingz=*sinx. (35) (i)Assumingasolution of~^-^+~5~z+~5~TQ^ke fthe formF(r+z),where r2=x2+y2+z2 ,obtain thefunction F;andby integrating withrespecttoz,deduce thesolution Fzlog(r-fz)-r..97 92F (li)Assumingasolution of"oT^^^i^^e^^efrm^(^)> where =#/\/, obtain thefunction</>;anddeduce asecond solution bydifferentiating withrespecttox.[London.] (36)Obtain arationalintegral function Vofx,y,zwhich satisfies thecondition 927927 and issuch astohave thevalue Az*atpoints onthesurface ofasphere ofunitradius with itscentre attheorigin. [Math. Trip.] MISCELLANEOUS EXAMPLES 235 (37)Show thatasolution ofLaplace's equation V2w==0 is u-(Acosnd4-Bsinn6)e***Jn(Xr), wherer,0,2arecylindrical co-ordinates andA,B,n,Aarearbitrary constants. [London.] (38)Show thatJn(r)(ancosn0+6nsinn0), where rand9are polar co-ordinates andanand fcnarearbitrary constants,isasolution oftheequation gay (39)Showhowtofindsolutions inseries oftheequation du9d2u"&- andsolvecompletelyforthecase inwhich, whenx=0, wa^-=Ccosh *.[London.] (40)Obtain twoindependentsolutions inascending powersofxof theequation d andprove bytransformingthevariables intheequation,orotherwise thatthecompletesolution maybewritten intheform whereAandBarearbitraryconstants.[London.] (41)Show thatthecompletesolution oftheequation where P,Q,Rarefunctions ofx,canbeobtained bythesubstitution y=yl+l/z )ifaparticular solution, yl9isknown. Show that,iftwoparticularsolutions yxandy2areknown, the completesolution is 1Rtot~ i)**+const - Obtain thecompletesolution oftheequation which hastwoparticular solutions, theproductofwhich isunity. (42)Show thatthedifferentialequation hasasolution oftheform(1+x)p(lx)q ,wherepandqaredeterminate constants. Solve theequation completely ;anddeduce, orprove otherwise, that if2aisapositive integer n,onesolution oftheequation isapolynomialinxofdegreen. [Londoa] 236 DIFFERENTIAL EQUATIONS (43)Verifythat 1-x*isaparticularsolution oftheequation a2 )t/~0, andsolve itcompletely. Bythemethod ofvariationofparametersorotherwise, solvecom pletely theequation obtained bywriting (1-a;2 )3instead ofzeroonthe right-handsideofthegiven equation. [London.] (44)Show thatthecompletesolution oftheequation where P,Qaregivenfunctions ofx,canbefound ifanysolution ofthe isknown. Hence, orotherwise, solve theequation d^y dy (45)Prove byputtingv***weixthat thecompletesolution ofthe d2v _dv .,, equationx-T-^-zn-j--fxv0, where nisaninteger,canbeexpressed intheform (Acosx-fBsinx)/(x)-f(Asinx-Bcosx) (x), where /(x)and(x)aresuitablepolynomials. [London.] (46)Ifu,varetwoindependentsolutions oftheequation where dashes denote differentiation withregardtox,provethat the completesolution isAu+Bv+Cw,where )f(x)dxfuf(x)dx andA,B,Carearbitraryconstants. Solve theequation x2(xa+5)y"/-x(7x +25)y/'+(22xl+40)y'~30x?/-0, which hassolutions oftheformxn . [London.] (47)Obtain twoindependent power-series which aresolutions of theequation.75 anddetermine theirregionofconvergence. [London.] (48)Prove thattheequation MISCELLANEOUS EXAMPLES 237 hastwointegrals , r- ._ where an"" \F(nTl)f"[London.] (49)Form thedifferential equation whoseprimitiveis Af. cosx\_/ sinx\y=*A(sin a;-\-- J+B (cosx-- J, where A,Barearbitrary constants. [London.] (50)Obtain thecondition thattheequation mayhaveanintegratingfactor which isafunction ofxalone, andapply theresult tointegrate (3xy-2ay2 )dx+(x*-2axy) dy-0. [London. ] (51)Show thattheequations dy^ have acommonprimitive, andfind it. [London.] (52)Prove thatanysolution oftheequation isanintegratingfactor oftheequation andconversely thatanysolution ofthelatterequationisanintegrating factor oftheformer. Henceintegratethe first ofthese equations completely,itbeing giventhat <p/P\RrT, , - [London-] (53)Iftheequation -|-fP-/-fQy==0,ax* dx wherePandQarefunctions ofx,admits ofasolution y=Asin(nx-f-a), whereAandaarearbitrary constants, findtherelation which connects PandQ., [London.] (54)Solve theequation ^|-4y2y t,d%>(1Xj having giventhat ithastwointegralsoftheform a+bx^M r., ysss^ [London.] 238 DIFFERENTIAL EQUATIONS (55)Show that thelinear differential equationwhose solutions are thesquaresofthose of-|+P(-^+Qy= (IOC dX maybewritten (J-+2p)(g+P| (56)Show thatthetotal differentialequation satisfies theconditions ofintegrability, andintegrateit. [London.] (57)Theoperatorj-being represented byD,show that ifXisa function ofxand(f>(D)arationalintegralfunction ofD, Extend theresult tothecase inwhichl/</>(D)isarational integral function ofD. Solve thedifferential equation d3v T-|+Sy=3x2+xe~2xcos sc. [London. ] (58)Show that 3-+4o-8y- hasanintegral which isapolynomialinx.Deduce thegeneralsolution. [Sheffield.] (59)Show that,ifintheequation Pdx+Qdy+Rdz=Q,PyQ,R arehomogeneousfunctions ofx,y,zofthesame degree,thenonevariable canbeseparated from theother two,andtheequation,ifintegrable, isthereby rendered exact. Integrate 23(x2dx+y2dy)+z{xyz*+z4-(x2+y2 )2 }(dx+dy) +(x+y){z*-z2(x2+y2 )-(x2+y2 )2 }dz-0, obtainingtheintegralinanalgebraicform. [London.] (60)Show that,iftheequation Pdx+Qdy+ Rdz=Qisexact, it canbereduced totheformAdu+/jidv=*Q;where X/Misafunction of u,vonlyandu=constant, v=constant aretwoindependentsolu- tions of dxdy.dz ^_a#~a#_ap~ap_aQ* dzdy dx dz dydx Hence, orotherwise, integrate theequation (yz+z2 )dx-xzdy +xydz=0. [London. ] (61)Prove thatz*~2xyisnotincluded in which isthegeneralsolution of {2y(*2-2xy)-2x-l}zp+{14-2y-2V(*a-2*y)}zq-x-y, butthat itisnevertheless asolution oftheequation. [Sheffield.] MISCELLANEOUS EXAMPLES 239 (62) (i)Showhowtoreduce Riccati's equation toalinearequationofthesecond order;andhence 01otherwise prove thatthecross-ratio ofanyfourintegralsisaconstant. (ii)Verify thatJ-fa;tanxyJ-xcotxareintegralsof anddeduce theprimitive. [London.] dx (63)Bysolving ^=-o>y, dy Jt=X intheordinary way,andeliminatingtfrom theresult, provethatthe point (x,y)liesonacircle. Alsoprovethisbyadding xtimes the firstequationtoytimes the second. [The equations givethevelocities, resolvedparalleltotheaxes, of point which isdescribingacircle withangular velocity o>.] (64)Find theorthogonal trajectoriesofthecurves y2 (a~x)~x8 . Prove thattheyreduce tothesystem r2=62(3-fcos20). [Sheffield.] dx (65) T-=ny-m, dy.J=fe-n*. dz whereI,m,nareconstants, provethat Ix+my-fnz, ,and areallconstant.Interpret these results. (66)Aplane curve issuch that thearea ofthetriangle PNT is mtimes thearea ofthesegment APN, wherePN istheordinate, NT thesubtangent atanypoint P,andAtheorigin, which isonthe curve; show that itsequationisyzm~l=a?m~2x. Show thatthevolume described bytherevolution ofthesegmentAPN about theaxis ofxbears aconstant ratio tothevolume ofthe cone generated bytherevolution ofthetriangle PNT. [London.] 240 DIFFERENTIAL EQUATIONS (67)Byusingthesubstitutions xrcos0, t/rsin0,orotherwise, solve thedifferentialequation Also findthesingular solution, andinterpretthe results geo- metrically. [London. ] (68)Show thattheequation canbereduced toClairaut's formbymaking y2-y?anewdependent variable;solve itandshow thatthesingular solution represents two rectangular hyperbolas. Verifyalso that this solution satisfies the given equation. [London.] (69)Prove thatthecurves inwhich theradius ofcurvature isequal tothelength intercepted onthenormal byafixed straightlineare either circles orcatenaries. [London.] (70)Solve theequation yx-%ap-fajP, andfindthesingular solution, givingadiagram. [London.] (71)Aplane curve issuch that itsradius ofcurvaturepiscon- nected withtheinterceptvonthenormal between thecurve andthe axis ofx,bytherelationpvc2 .Show that,iftheconcavityofthe curve isturned away from theaxis ofx, t/2c2sin2 </>+&, where <f>istheinclination ofthetangenttoOx. Obtain thevalue of a?asafunction of <f>inthecase 6= ;andsketch theshapeofthe curve. [London.] (72)Show that,ifthedifferential equationofafamilyofcurves be giveninbipolar co-ordinatesr,r',9,9',thedifferential equationofthe orthogonal trajectoriesisfound bywriting rd9 fordr,r'dfffordr', -dr forrd9, -dr' forr'dQ'. Find theorthogonal trajectoriesofthecurves ab -r+?~C> cbeing thevariableparameter. [London.] (73)Thenormal atapointPof-acurve meets afixedstraightline atthepoint G,andthelocus ofthemiddlepointofPG isastraight lineinclined tothefixedstraightlineatanangle cot^S. Show that thelocus ofPisaparabola.' [London.] (74)Solve theequation 2(p-l)y=*pzx;show that the"^-dis- criminant" isasolution oftheequation,and istheenvelopeofthe familyofcurves given bythegeneralsolution. [London.] (75)Obtain thedifferentialequationoftheinvolutes oftheparabola y1iax,andintegrateit.What isthenature ofthesingularsolution 1 [London.] MISCELLANEOUS EXAMPLES 241 (76)Prove that ifthenormals toasurface allmeet afixed straight line,thesurface must beoneofrevolution, [London.] (77)Integratethepartialdifferential equation Give thegeometrical interpretationofthesubsidiary integrals and ofthegeneral integral. [London.] (78)Integratethedifferential equation z(*+2^)~-*(y + 2z)|==t,*-*. Find theparticularsolutions sucn that thesection byanyplane paralleltoz=shall be(i)acircle, (ii)arectangular hyperbola. [London.] (79)Afamilyofcurves isrepresented bytheequations where a,/3areparameters. Prove thatthefamilyofcurves canbecutorthogonally byafamily ofsurfaces, andfindtheequationofthisfamily. [London.] (80)Solveb(bcy+axz)p+a(acx+byz)q=ab(z2-c?) i andshow that thesolutionrepresents anysurfacegenerated bylinea meeting twogivenlines. (81) (i)Solve L where L,R,andEareconstants. [Thisistheequationfortheelectric current /inawire ofresistance Randcoefficient ofself-induction L,under aconstant voltage E.] (ii)Determine thevalue ofthearbitraryconstant if/=/ when t-0. (iii)Towhat value does /approximate when tislargeI [Ohm's lawforsteady currents.] (82)Solve L~+RI~Ecospt. [Thesymbols have thesamemeaningasinthelastquestion, except that thevoltage Ecosptisnowperiodicinstead ofbeingconstant. Thecomplementaryfunction soon becomesnegligible,i.e.the free oscillations ofthecurrent aredamped out.] (83)Find theParticularIntegralof [This givesthecharge QononeofthecoatingsofaLeyden jar when aperiodicelectromotive forceEcosptacts inthecircuit con- necting thecoatings. TheParticularIntegral gives thechargeafter thefreeelectrical oscillations havebeendamped out.] 242 DIFFERENTIAL EQUATIONS (8d)Show thattheequations dx dy dx dt dt dt aresatisfied bythe trial solution y=mx,provided that inisaroot of thequadratic 2+3w 16+3w dxandxisgiven by7-=--(2-f 3w)z=0. Henceprovethattwosetsofsolutions ofthedifferential equations arey=x and y- sothatthegeneralsolution isx=Ae2i+Be~l , (85)Usethemethod ofthelastexampletosolve dzx 7-+23^-8^=0, [Equationsofthistype occur inproblems onthesmall oscillations ofsystemswithtwodegreesoffreedom. Themotion given byy=%x (orbyt/==-5$)issaid tobeaPrincipalorNormal Mode ofVibration. Clearlyitissuch that allpartsofthesystemaremoving harmonically with thesame period andinthesamephase.Ify-2xandy+5o5are taken asnewvariables instead ofxandy,theyarecalledPrincipalor NormalCoordinates.] (86)Given thatL,M,N9R,Sarepositive numbers, such thatLN isgreater thanM2 ,provethatxandy,defined by diminishindefinitely astincreases. [Show thatx=*Aeat+Bebtandy=>Eeat4-Febt ,where aand 6are reaZandnegative. Theseequations givethefree oscillations oftwo mutually influencing electric circuits. LandNarecoefficients of self-induction, Mofmutual induction, andRandSareresistances.] (87)Show (without working outthesolutions infull) that the Particular Integralsofthesimultaneous equations MISCELLANEOUS EXAMPLES 243 areunaltered ifinthe firstequationtheterm I-dtisomitted andL isreplaced byL-- ^. [This follows atoncefrom thefactthattheParticularIntegralsare oftheformAsin(pt-a). Theseequations givethecurrents intwomutually influencing circuits when theprimary, which contains acondenser ofcapacity c, isacted upon byanalternatingelectromotive force. Thisexample shows that theeffect ofthecondenser canbecompensatedforbyin- creasing theself-induction.] <88>If i and M a whereLN-M2isaverysmallpositive quantity,show thattheCom- plementaryFunction forxrepresentsavery rapidoscillation. [These equationsoccur inRayleigh's theoryoftheoscillatorydis- chargeofacondenser intheprimarycircuit ofaninduction coilwith aclosed secondary. Notice thatthesecond equation shows that the secondary current isatitsmaximum when theprimarycurrent isatiti minimum. SeeGray's Magnetism andElectricity,Arts. 489and490.] (89)Prove thattheParticularIntegralsofthesimultaneousequations d2xm -a(x-X)+kcospt, u.maybewritten x=-^^cospt, -ak where 6=mp2-aandB=Mp2-(a+A). Hence show thatxandXareboth infinite fortwospecial values ofp. [These equations givetheoscillations ofthe"elastic doublepen- dulum." MassesmandMarearrangedsothattheycanonlymove inthesame horizontal line.AspringconnectsMtoafixedpointof this lineandanotherspringconnects mtoM.Aperiodicforce acts upon m,andthesolution shows thatbothmasses execute forced vibra- tions whose amplitude becomes very largefortwospecialvalues ofp. Ofcourse this isthephenomenonofResonance again.Itisimportant tonotice that thevalues ofpthat giveresonance inthiscase arenot thesame astheywould beifonlyonemass werepresent.Thismay beappliedtothediscussion ofthe" whirling"inaturbine shaft. SeeStodola's SteamTurbine.] 244 DIFFERENTIAL EQUATIONS (90)Show thatthesolution ofthesimultaneousequations wherem=*M anda=6,maybeexpressed bysayingthat6and <f>are eachcomposedoftwosimple harmonic oscillations ofperiods %7r/pi and %Tr/p z,p^andpa2beingtheroots ofthequadraticinp2 , 28aV-Slagp2+2702=0. [These equations givetheinclinations tothevertical oftworods ofmassesmandMandlengths2aand2brespectively when theyare swinginginavertical planeasadouble pendulum, thefirstbeing freely suspended from afixedpoint andthesecond from thebottom ofthe first. Thetwo oscillations referred toareknown asthePrincipal (or Normal) Oscillations. Similar equations occur inmany problemson small oscillations. Adetailed discussion ofthese isgiveninRouth's Advanced Rigid Dynamics, withspecialreference tothecasewhen the equationinphasequal roots.] <91> +*-* [These equations give themotion ofthebob ofagyrostatic pen- dulum which does notswingfarfrom thevertical. Notice that ifthe initial conditions aresuch thatBQ,wegetmotion inacircle with angular velocity p,while ifA=0,wegetmotion inacircle withangular velocity qintheoppositesense. (For p,q,A,Bseetheanswers.) Similar equations hold forthepath ofrevolvingions intheex- planationoftheZeeman Effect (the trebling ofaline inaspectrum byamagnetic field). SeeGray's Magnetism andElectricity,Arts. 565-569.] (92)Given (dx dz , 2-** where o,6,eareconstants, obtain adifferential equationforz. Hence provethat ifz=-=-when t0, z-c-f-^T [be~at-ae~bf ]. [These equationsoccur inPhysical Chemistry when asubstance A forms anintermediate substance B,which then changesintoathird MISCELLANEOUS EXAMPLES 245 substance C. x,y,zarethe"concentrations"ofA,B,Grespectively atanytime t.SeeHarcourt andEsson, Phil. Trans. 1866and1867.] (93)The effect onasimple dynamical system withonedegreeof freedom ofanyother dynamical systemtowhich itislinked canbe represented bytheequation Iftheexciting systemofwaves ismaintained steadysothat XAcospt,findthevalue ofpforwhich there isresonance, andprove that ifJULexceeds acertain value there isnoresonance. Draw curves illustratingboth cases. [Math. Trip.] (94)Solve thedifferential equation +2/be+ti2z=when ft*<n2 . Inthecase ofapendulum makingsmall oscillations, thetime ofa completeoscillation being2sees,andtheangular retardation dueto theairbeing taken as-04x(angular velocityofpendulum), show that anamplitudeof1will in10complete oscillations bereduced toabout 40'. [Take log,e=4343.] [Math. Trip.] (95)Themotion ofasystem depends practicallyonasingleco- ordinate x;itsenergyatanyinstant isexpressed bytheformula \mx2+\ex2 ;andthetime-rate offrictional dampingofitsenergyis \kx2 .Prove that theperiod (TO)ofitsfreeoscillation is 1&2\-4 Prove thattheforced oscillation sustained byadisturbingforce of ek2 typeAcosptisatitsgreatest whenp2=--r^andthattheamplitude M 1l ofthisoscillation isthen~ 9while itsphase lagsbehind that ofthe 7T/C forcebytheamount tan"1-.[Math. Trip.] (96)Show thatthesubstitution T-- (-=- Jreduces i7*+K:jf)a -Q TT/TJ tothelinear form-j--f2PT=Q. From vriththeconditions^-=0ands=*2awhenJ0, obtain -*- , <Psgud -. 246 DIFFERENTIAL EQUATIONS [This givesthesolution ofthedynamical problem:"Auniforn chain iscoiled uponahorizontalplane andoneendpasses over fi smoothlight pulleyataheightaabove theplane ;initiallyalengtl 2ahangs freely ontheother side. Prove thatthemotion isuniforml) accelerated." SeeLoney's Dynamics ofaParticle andofRigid Bodies p.131.] (97)Find asolution oftheequation 3/290\1d(..30\x- 1r2-^- )+ .K^ \sin6=1=0 or\or/ sin606\ dd/ oftheform <f>=/(r)cos0, given that ~=Fcoswhen ra and-S-=awhen r=oo. dr [<^>isthevelocity-potential when asphereofradius amoves witl velocity Finastraightlinethrough aliquidatrestatinfinity.Se< Ramsey's Hydro- Mechanics, Part II.p.152.] (98)Find asolution of -.=c2~ fjt Oli which shall vanish when x=0,andreduce toAcos(pt+a)when x=5. [This givestheform ofoneportionofastretchedstring,fixed ai both ends, ofwhich agiven pointismade tomove with theperiodi< displacement Acos(pt-ha).Theportion considered isthatbetween th< given point andone oftheends. SeeRamsey's Hydro-Mechanics Part II.p.312.] (99)Obtain thesolution of \3>a+rdr, intheformr<f>*=*f(ct-r)+F(ct+r). [<f>isthevelocity-potentialofasphericalsource ofsound inair SeeRamsey, p.345.] (100) Obtain asolution of such thatd(/>ldy=*Qwhen y-h and varies as coB(mx-nt) when#=0. [</>isthevelocity-potentialofwaves inacanal ofdepth h,thesidoi being vertical. SeeRamsey, p.265.] (101) Obtain thesolution ofthesimultaneous differentialequations dx MISCELLANEOUS EXAMPLES 247 with theinitial conditions ^dx _ dii _,-, y-O. -0, =0, mtheform z= where z=x+iyandg= Show thatthesolutionrepresentsahypocycloid contained between twoconcentric circles ofradiiaandan/q. [This example givesthetheoryofFoucault's pendulum experiment demonstratingtherotation oftheearth. SeeBromwich, Proc. London Math. Soc. 1914.] (102) Obtain anapproximatesolution ofEinstein's equationof planetary motion $um inthefollowing manner : (a)Neglectthesmall term3mu2 ,andhence obtain w=S{l+ecos(<f>- ui)},asinNewtonian dynamics. (6)Substitute thisvalue ofuinthesmall term3mu2 ,andhence obtain d?u m3m36m3 . 3m3e2 dfi+U=W+~W+p-ecos(0-o) +^^{ (c)Neglectalltheterms ontheright-handside ofthis differential equation except,-2and-7^-ecos(0 or).Theterm incos(0- trr)must beretained;itisofthesameperiodasthecomplementary function, and thereforeproducesacontinually increasing particular integral. [Seethe resonance problemEx.36onp.46.]Hence obtain u= 2\1+ecos(0-or)+-p-e<f>sin(0-or)h =,2{1+ecos(0-or-e)}approximately, where e=-y^-and e2isneglected. [Thisresultprovesthatwhen theplanetmovesthrough onerevolu- tion theperihelion (givenby0-CT-e=0)advances afraction ofa revolution givenby-=-^-.When numerical values aregiventothe constants itisfound that Einstein's theoryremoves awell-known discrepancybetween observed and calculated results onthemotion oftheperihelionofMercury.SeeEddington, ReportontheRelativity Theory ofGravitation, pp.48-62.] 248 DIFFERENTIAL EQUATIONS (103) L(x, y,x',y')isafunction ofthevariables x,y,x',y*. X,Yaredefined bytheequations T3LvdL *~M' *~dy'' Ifthese equations canbesolved forx'andy'asfunctions ofX,Y,x,y, and ifH(X,Y,x,y)isthefunction obtained byexpressing Xx'+Yy'-L entirelyinterms ofX,Y,x,y,thenprovethat dn dLand~___ .^j Prove alsothattheequation dt\dx'J dx istransformed into -rr=-^~ (4)dt dx^' [ThisistheHamiltoniantransformationindynamics. Equation (3) isatypical Lagrangian equation ofmotion ingeneralised co-ordinates. Hamiltonreplacesitbythepairofequations (1)and(4). SeeRouth's Elementary Rigid Dynamics, Chap.VIII. This transformation should becomparedwith that ofEx.21ofthemiscellaneous setattheendof Chap. XII., where wehadtwopartialdifferentialequations derivable fromeach otherbythePrinciple ofDuality.] (104)Show that Jacobi's method(Art. 140)appliedtoHamilton** partial differential equation s-+a(xltX2,...xnyPitp2,pntt)**Q leads to ~~r~~ %>~j~~~ a~~"(f^l* 2,...n), which aretheequationsofmotion ofadynamical system,inHamilton*! form.[SeeWhittaker's Anal;/fical Dynamics, 2nd ed.,Art.142.] (105) (i)Prove that if u(x ty,z)=*a and v(x, y,z)^b areanytwointegralsofthesystemofdifferentialequations dxdydz P(X, y,z)" q(x, y,z)" r(x, y,z)9 . 1d(u, v)1d(u, v)Id(u, v). . then-^7r-^7-r=- ,^7r=m(x, t/,z),say, pd(y>z)yo(z>x)rv(x,y) [tniscalled amultiplierofthesystem.] MISCELLANEOUS EXAMPLES 249 (ii)Show thatmsatisfies thepartialdifferentialequation (iii)Ifn(x yy9z)isanyothermultiplierofthesystem, show that d andhence that -^7 ~~^=0 identically, d(x, y,z)JJ sothatm/nisafunction ofuandv,anlmjncisanintegralofthe original systemofdifferentialequations. (iv)Ifu(Xj y,z)acanbesolved forz,givingzf(x9y,a),and ifcapitalletters F,P,Q,R,Mdenote thefunctions ofx9y,a,obtained bysubstitutingthisvalue ofzinv9p9q9r,m,then provethat dx dij V(x 9y,a)=6isanintegralof-p= ^. Prove alsothat 3fP= -~-^~dyoz amdvdu and MQ= ^~or / d \ (where^istobeexpressedinterms ofx9y9a\sothat [This suggeststhat ifanyintegral w=oandanymultiplier mare known, thenM(Qdx-Pdy) l~ willbeaperfect differential, leading toanintegralofthesystemwhen aisreplaced byu(x yy9z). Foraproofofthistheorem seeWhittaker's Analytical Dynamics, 2nd ed.,Art. 119.Amoregeneraltheorem isthat if(n-1) integrals ofasystemofdifferential equations dx1dx2dxn__dx Ih**!^**'"***!**" P areknown andalsoanymultiplier,thenanother integralcanbedeter- mined. This isgenerallyreferred toasthetheorem ofJacobi's Last Multiplier.InDynamics,where thistheorem isofsome importance (seeWbittaker, Chap. X.),thelastmultiplierisunity.] (v)Show thatunityisamultiplierof dx dydz anda^+t^+z^a anintegral, sayu(x 9y9 Show that inthiscase andhence obtain thesecorjcj integral DIFFERENTIAL EQUATIONS ibe*'/(0 dt,where aand6areconstants, then a Henceprovethatywillsatisfy thedifferential equation if and Usethismethod toobtain asasolution, validwhenx>0, of ^V dy . sJ+-*-a Thecorrespondingsolution forthecasesc<0 isobtained bytaking thelimits ofthe firstintegralas1toco ,instead of-co to-1. [Exs.106408givesome ofthemostimportantmethods ofobtaining solutions ofdifferentialequationsintheform ofdefiniteintegrals.] (107) Verify that v=v+~-\ e~*dz VTTJo ,,. . dv d*v isasolution of^-=A:^~o>dt dx? reducing, when t=0,toVQ+Vforallpositivevalues ofxand tov-V forallnegativevalues. [visthetemperatureattime tofapointatadistance xfrom a certain planeofasolidextendingtoinfinityinalldirections, onthe suppositionthatinitiallythetemperaturehadthetwodifferent constant values v+VandVQ-Vonthetwosides oftheplanex=0. Kelvin used thisexpressionforvinhisestimate oftheageofthe earth(seeAppendix DofThomson andTait's Natural Philosophy). The discoverythatheat iscontinually generated bytheradio-active dis- integrationoftherocks introduces anewcomplexityintotheproblem.] (108) (a)Show that F=ffe'*+w*+ne/(s,t)dsdt (thelimitsbeing anyarbitrary quantities independentofx,t/,z)isa solution ofthe linearpartialdifferential equationwith constant coefficients 3 MISCELLANEOUS EXAMPLES 251 ifltm,nareanyconstants orfunctions ofsand tsuch that F(l,m,n)=0. Extend thetheorem tothecasewhen there arenindependent variables x,y,z,...,and(n- 1)parameters s,t,... . Obtain F- [[e*<*cost+y8lnt+*z)/(s, )<fo(ft ,.. 92F92FdV 32Y asasolution of -=-+Vr=-^~- [H.Todd.]3x23/2ozL J (O*J ~\\ p"~5~~jr)^=^8ahomogeneouslinear partialdifferential equationwith constant coefficients asolution is 2, )e&, where thelimits areanyarbitrary quantities independentofx,y,z,and /,mynareanyconstants orfunctions oftsuch that F(l,m, n)=0. Extend thetheorem tothecasewhen there arenindependent variables and(n-2)parameters. [SeeH.Todd, Messenger ofMathe- matics, 1914.] Obtain F= Jf(xcos t+ysin 14-iz, t)dt 32F92F92FA%sasolution of 19+ ~^~9+~^~r""0. d^2 a?/2^2 [Whittaker'ssolution ofLaplace's equation.] (109)Bysubstitutingthetrial solution a,a.-ao+x+^+- inthedifferential equation -,- -I-y=- ,ax x . .t0! 1! 2! 3! obtain theseriest/=++-^+~4+ - Prove that thisseries isdivergentforallvalues ofat. Obtain theparticular integral exfx ex -r J-00* andbyrepeated integration bypartsshow that e*,0! 1! 2! n! f*(n+!)! Hence provethat ifxisnegativetheerror obtained bytakingn-f1 terms oftheseries instead oftheparticular integralislessthan the numerical value ofthe(n+l)thterm. [Suchaseries iscalled asymptotic.SeeBromwich's Infinite Seriet, Arts. 130-139;or2nd ed.,Arts. 106-118.] 252 DIFFERENTIAL EQUATIONS (110)Show that ifthesequenceoffunctions fn(x)bedefined by /(x)a-fb(x-c),where a,6,careconstants, and, * Hence show that*/=n(x)^9asolution of provided that certain operations with infinite series arelegitimate (foi aproofofwhich seeWhittaker andWatson's ModernAnalysis, p.189.* They giveaproofoftheexistence theorem forlinear differentialequa- tions ofthesecond orderbythismethod). (Ill) Prove thatthesolution fofthetwosimultaneous linear differ- entialequations withconstant coefficients (whereDstands ford/dt),maybewritten whereVisthecomplete primitiveof Hence show that ifthedegrees of/, .F,</>,\fsinDbep,qyr,8respec- tively, thenumber ofarbitrary constantsoccurringinthesolution will ingeneral bethegreaterofthenumbers(p+s)and(q+r),but if (p+s)=(q+r)thenumber ofarbitrary constants maybesmaller, and mayevenbezero asintheequations (112) (a)Prove that if-u(s), y~v(x) areanytwosolutions ofthelinear differentialequation ofthefirstorder then(vMj sothatvau,where aisaconstant. (b)Prove that ify=u(x), y**v(x), y**w(x) *p.195in3rdand4theditions. tThis,jt maybeproved, cannot bethemost general solution ifitgives thenumber ofdifferentarbitrary constants forxandytogether lessthan forF,a*will happeniff(D)andF(D)have acommon factor other thanamere constant. MISCELLANEOUS EXAMPLES 253 areanythree solutions ofthelinear differential equation ofthesecond order P() then P-j-(WJL-vwj)+Q(wv l-wo^=0 ctx and Pj-(uv 1-vttl)-\-Q(uv 1-vul)=0. Hence show that w=au+bv. [Byproceeding stepbystepinthismanner wemayshow that a differential equationofsimilar form butofthewthorder cannot have more thannlinearly independent integrals.] (113) Letw,v,wbeanythree functions ofx. Prove that ifconstants a,b,ccanbefound sothaty~ vanishes identically, then uvw W =0, while conversely,ifthisdeterminant(theWronskian) vanishes, the functions arenotlinearly independent. Extend these results tothecase ofnfunctions. [Considerthedifferential equationofthesecond order formed by replacing u,u^,u2inthedeterminant byy,yvyzrespectively. Such anequationcannot havemore thantwolinearly independent integrals. TheWronskian isnamed afterHoene Wronski, oneoftheearly writers ondeterminants.] (114) Prove that z=e^x^"1^satisfies thepartialdifferential equation Hence,ifJn(x)isdefined asthecoefficient oftnintheexpansion eW'-VO-f}^*), oo provethaty~J n(x)satisfies Bessel's equationoforder n, [The operations with infinite seriesrequire some consideration.] (115)Ifuxdenotes afunction ofx,andEtheoperatorwhichchanges uxintoux+l,provethefollowingresults : (i)Eax=a.a*, i.e.(E-a)a*=Q. (ii)E2aa5=as.a*. (iii)E(xax )=a(xax )+a.a*,i.e.(E-a) (xa*)=a.a. (iv)(E-a)*(xa*)=Q. (v)(pQE2 -\'plE+p^aai=(p^'}-p la+p 2)aa> 9ifthep'sareconstant. P.D.B. S 254 DIFFERENTIAL EQUATIONS (vi)ug!**Aai*+Bb* isasolution ofthelineardifference equation i.e. if^4and J9arearbitrary constants andaand6theroots oftheauxiliary equation pw?+p lm4-p2=0.(Of.Art.25.) Solvebythismethod (2#2+5^+2)^=0. (vii)utt*=(A+Bx)a*isasolution of(E2-2aE+a?)ux=Q. Here theauxiliary equation w2-2aw+a2=0hasequalroots. (Cf.Art.34.) (viii)ux=r*(P cosxQ+QsinxO)isasolution of ifPandQarearbitrary constants, piqtheroots oftheauxiliary equation pQm2-fjjjW+p%= andp+t'2=r(cos -ftsin0). (Cf.Art.26.) Solve bythismethod (E2 (ix)Thegeneral solution ofalinear differenceequation withconstant coefficients isthesum ofaParticularIntegral andtheComplementary Function, thelatterbeingthesolution oftheequationobtained bysubstituting zero forthefunction ofxoccurring ontheright-handside.(Cf. Ar.29.) (x)ax/F(a)isaparticular integralof provided thatF(a)^0. (Cf.Art. 35.) Solve bythismethod (E*+8E-$)u x=*2*. [For furtheranalogies between differenceequations and differential equations, seeBoole's FiniteDifferences, Chap.XL] (116)Show thatbyapplyingthemethod ofArt.53toLagrange'g equation y~xF(p)+f(p), wegetingeneral (butnotforClairaut's form, where F(p)=p)thecom- plete primitiveintheparametric form Hence show that ifCv(72,(73areanythree curves included inthis primitive, correspondingtothevalues cltc2,c3ofc,andPI(XVt/j), P*(X2>!/2)>^3(^3* 2/3)pointsonCltC2,C3respectively,such thatthe tangentsatthesepointsareallparallel,then MISCELLANEOUS EXAMPLES 255 i.ePj,P2,P3are collinear, andtheratioPxP 3P%P$isconstant asthepoints move, eachalongitsowncurve, insuchaway astokeep thecorresponding tangents parallel. [Thus giventwocurves included inthecomplete primitive, wecanconstruct geometrically anynumber ofothers.] (117) Prove thataplane curve, suchthatthelengthoftheradius of curvature atanypointistwice thelengthofthenormalintercepted between thecurve andafixedstraight line,iseither acycloid, whose base isthestraight line, oraparabola,whose directrix isthestraight line. [London.] (118)Acurvepossessesthepropertypkt&n^wherepisthe radius ofcurvature, \]sistheanglethetangent makes withtheaxisofxt andkispositive. Show that thecurve hasabranch given bythe equations x=k(l-cos0),y=&{lo(sec6+tan6)-sin0}, where 0:S0<TT, andtheoriginistaken atthepoint0=0. Show that, ifsisthelengthofthearcmeasuredalongthisbranch from thesame point, k s=klogf.[London.]KX (119) Obtain asolution oftheequation -~a=*c2^~2*n^e *orm f(x)sinmt,which issuch thatx --*=K9aconstant, whenx=0andJ=0, ot ~-0, whenx=0, forallvalues oft. [London.]ox X-V.O O9 (120) Obtain fortheequation ;r-2+~~"2asolution which satisfies thefollowing conditions :^ (i)when y=0,z=sinx\ (ii)whenx~0 orx,z; (iii)zdoesnotbecome infinite anywhereintheregionoftheplane ofxyyinwhichy>and TT>x>0. [London.] (121)Bytwointegrations bypartsshow that,ifP,Q,Rarefunctions ofx,andsuffixes denote differentiations withrespecttox, +Ry)dx-z(PVl+Qy)-y(Pz) l+y{(P*) 2-(Qz^+Bztfx. Deduce thatthetwoequations aresuch thatanyintegralofone isanintegratingfactor oftheother. [Such equations aresaidtobeadjointtoeachother.] 256 DIFFERENTIAL EQUATIONS Show that,ifDrepresentstheoperator djdxttheequation adjointto is{D-q(x)}{D-p(x)}z=0. Verifythis fortheequation ya+(+x2)y14-(2x+x3)y=0. [Here J>(s)=a, q(x)=x*.] General solution of;rl=4 !*!-9x2a2dt* Factorisingtheoperator,theequation maybewritten /313U/3 19\ In_y913MY3 *9> i\ \fo~adiJ {\dx+aWy]-""^dx+aWt)\\dx~a~di)y)' Hence(cf.p.33)theoriginal equationissatisfied byanyintegral ofeither ofthetwoLagrangelinear equations S4&-0 andJM&-OL oxaot dxaat Forthe first ofthese thesubsidiary equationsare(from Art.123) dx dt_<fy T~I/a~o~' Twoindependent integrals are y=6,x-at=*o. Thegeneral integralis -/(*-aO- Similarly thesecond Lagrange equation gives y=JF(a& +a^).These arebothintegralsoftheoriginaldifferentialequation. Asitislinear, athirdintegralis containing twoarbitrary functions, andnomoregeneral solution ofan equationofordertwocanbeexpected. (Cf.pp.61and218.)Asimilar method canfoe used fortheequationofArt. 145. TheMethod ofParameters. (C.N.Srinivasiengar.) Ifapartial differential equation becomes anidentity onsubstituting p=/(x, a)/(p(z, a),q^F(y, a)/<f>(z9a),wecanusetheseexpressionsin conjunctionwithdz*=pdx +qdytoobtain thecomplete integral \<f>(z,a)dz\f(x, a)dx+\F(y, a)dy+b. Forexample, theequationz2(p+q)=x*-fyabecomes anidentityif y=(a;a+a)/s2 ,q=(y2-a)/z2 , givingz*x8+y8-fSax-3ay+6. Thismethod willdealwith allequationsofStandard Forms Iand III(Arts,f29and131)andsome ofII(Art. 130). ANSWERS TOTHEEXAMPLES CHAPTER I. Art. 5. . . (5)Thetangent toacircle isperpendiculartothe linejoiningthe pointofcontact tothecentre. (6)Thetangentatanypointisthestraightline itself. (7)Thecurvature iszero. Art. 8. Z S A (1)y (2)ya-fbx-a^-6^+a--=+...=acosx+6sinx. 41 Ol 4:1 Miscellaneous Examples onChapter I. {'+(!)'}:-<:)' t- (12)y-ae+6ar.(14)60and-60 U DIFFERENTIAL EQUATIONS d?y (15) Differentiate andput&1,y=*2.Thisgivesj--2andhence/>, (17) (i)aj+10; (ii) 2/2 CHAPTER II. Art. 14. (1)6a^+5x!/+^2~9x-42/=o. (2)sinxtany-f sin(sc (3)secajtany-ea5o.(4)&-t/+c=log(z+ (5)x+ye****cy. (6)t/=cz. (7)ey(sinx+cosaj)=-c. (8)ofy+4c?/-f4=0. (9)ye'cB.(10)sinxcosyo. Art. 17. (1)(x+y)*~c(x-y). (2)z2+2t/2 (c-flog y) (3)xf-c(x-y)*. (4) (5)(2x-?/)2=c(a;+2?/-5). (6) (7)s-y+c-log(&j-4y +l). (8) Art. 21. (1)22/==(a; +a)8+2c(-fa)8 .(2)a?y=sinx-fccosx. (3)7/logx=(logx)2+o.(4)a?3= ?/3(38ina; +c). (5)^(s+ce*)-!; (6)x2/3 -f-ci/. (7)-e-"(c+tant/), Art. 22. (1)Theparabola y2=4ase+c. (2)Therectangular hyperbola xy=c2 . (3)Thelernniscate ofBernoulli r2=a2sin20. xc (4)Thecatenary y=A;cosh-j.(5)xy^c2 . (6)y'-ai' +o1 .(7)yP=cafl.(8)r2=ce". (9)logr+J024-J08-o.(10)Theequiangular spiralsr=ce9ten Miscellaneous Examples onChapter II. (1)xy=f+c.(2) (3)sinasiny+esln*<>.(4) (5)cxy=3y+'v/(ya-xa ). (11) (12)tm~l (xy)+\og(x/y)=*c. (14) (x2-] (15) (i)TheReciprocal Spiral r(0-a)o. (ii)TheSpiralofArchimedes r=c(0-a). (16)Theparabola 3Ay2~2s.(18)a=y(o-4logy). ANSWERS 111 (19) (i)a+(y-c)a1+c*,asystemofcoaxal circles cuttingthegiven system orthogonally, (ii)r2ce~e\(iii)n2=>r{c+log(cosec nO 4-cotn6)}. (20) (21) CHAPTER III. Art. 28. (1)y-Aer9+Be~**. (2)y4cos2a?+Bsin 2a?. (3) t/-4-8a8+J5e~4*. (4)ye2x (,4cosx+/^Binx). (5)--(4cos3< +J3sin30. (6)*- (7)y-4e +Be-+06-a .(8)y=2 (9)y-^cos(2iC-a)+JBcos(3x-^). (10)y-4cosh(2x-a)H-/*cosh(3.i;-$),or (11)y (12)y-A&*+Be~2x+Ee~xcos(x^/3-a)4-.Fc*cos (13)=acos (16)g-0e-oosn +~8inn, where n- Art. 29. (1)y-e^l+^cosx +Bsinz). (2) (3) !/=2sm3x-f ^1cos2x+sin2z.(4)a=2;6=1. (5)a=6;6--1.(6)a--4;p=2.(7)o=l;6-2; (8)a-2. (9)4^*.(10)3e7iC . (11)-fsin5s. (12)flcos5a-jJsin5-c. (13) 2. Art.34. (1)y (2)y (3)y=(A-f-j&e)e*+Ecos#4-JFsinx. (4)y- Art.35. (1)y-2e8a!-fe-8a!(^co84x +Bsin4x). (2)y=e~v*(A cosjj?-fBsinjs)4-eaa! /{(a (3) ?/= (4)y- IV DIFFERENTIAL EQUATIONS (5)y=(A4-ax/2p) coshpx+Bsinhpx. (6)y Art.36. (1)y=2sin2a;-4cos2x +.4e-"*. (2)y=4cos4z-2sin4z+^le2*+Be3*. (3)y=2cos a?+e~4x (^4cosSo;+BsinSo;). (4)y-sin20z+e-*(^fcos20z 4-5sin20z). Art.37. (1)y-a?-3a5* +63--6+-4e-.(2)y-6s2-6s4-4 (3)y (4)y (5)y (6)y Art.38. (1)y=4cosa +(B+2z)sma;. (2) (3)y-Ae2x (4)^={^1sinic-f(B--a;) cosx}e~x . (5)y^(44-Bx-a;3 )cosx+(E+Fx+3x2 )sinx. (6) ?/ (7)y={^Isin4o?+(B-x+x2 )cos4z}e3x . Art. 39. (1)y~Ax+Bx*+2x*. (2)y2+^4x~4cos(3logx)+Bx~* sin(3logz). (3)y=8cos(logx)-sin(log x)+J.or2+I?xcos(v"3log05-a). (4)y=4+logx+Ax+Bxlogx+Gx(logx)2-fDo;(logx)3 , (5)y(l+23)2 [{log (1-f2z)}2+Alog(1+2z)+B], (6)y^cos{log (1+B)-a}+2log(1+)sinlog(1+x). Art. 40. (1)y*=*Acos(o3-a) ;as-Asin(#-a). (2)y-A<*x+Be3a: ;-64e5x-7Be3aj . (3)y^e^H-Bcos(2x-a) ;z=2Aex-Bcos(2x-a). (4)y=6^+4+Be~2*;*=e*+4-Be-2a5 . (5)y=Acos(o;-a) +4Bcos (2x- /3)+cosTo;; z=*A cos(a;-a)4-Bcos(2x- /3)-2cos7x. (6)y-S^e335-4Be4*+2e-a+cos2x-sin2a;; z-4e8aj-fBe4* -I-3e-*+4cos2x-f6sin2x. ANSWERS Miscellaneous Examples onChapter IIL (1)y~(A+Bx+Cx*)ex+2e*x .(2)y=(A+Bx+Gx*) (3)y=Ae~**+Be-**+Ce~x+E+2e-2*(sinx-2cosx). (4)y4e* -fcos(2x~a)-2e*(4sin2x+cos2x). (5)y=(A+Bx+Cx*)e~* +(E+x+2x*)e**. (6)y=Asin(x-a)+Bsinh(3x- /3)-2sinh 2x. (7)y~(A+Bx+5z2 )cosh a;+(E+Fx)sinh x. (8)^3+4z+2z2+(4+J3x+4z2)e2a5icos2x. (9)y=(A+j5x-f3sin2x-xcos2x-2x2sin2x)e2*. (10)y=Acos($-a)+|-^cos2x-\xcosx+TVsin3x. (11)y^4cos(a?-a)4-J5cos(3x~/3)~3xcosx-fxcos3x. (12)y (13)y-J-f-J51ogx-}-2(logx)3 . (14)y- (15) i/ (16)y (17)x-^e8'-f<r3' -IEcos t-fJ?sin-e*; t/Ae425^e-3+(3JS-4J)cos (+(3F+4fl)sin <-e (18)x=^62t+J5e-fcos(V^-a); t/-Ae* -fJ5e~'cos(V&-a-f2?r/3) ; ^4e2f+fie-4cos(\/3i-H-4-7T/3). (19)x**At+Btrl ;y~Brl-At. (20)x==4cos(logt-a)4-J^"1cos(logJ- ); y=Atsin(logJ-a)-Bt~lsin(logt- /3). (27) (i)(z-l)e2*; (ii)J(x2-2x+l)sinx-fJ(xa-l)cos*. (31)y-^-f 4e*. (32)y=(sinax)/(p2-a?)+A cospx+Bsinyx. (33) ?/^eaa!+Beb+e&xe^a"^x (logx-1)dx. (35) (iii)y^Acos(x-a)xcosx4-sinxlogsinx. (37) (i)Jc/(2phe);(ii)zero. (38)y=JBcosnxi-Fsinwx+coshMX4-1/sinhnx. dz dz (1)aTflS-CHAPTER IV. Art. 42. (2)^"2+5~a=a''(Laplace's equationintwodimensions.) /o\"^ "^ i.C/Zi.. v^I C/S _ vXvVttvt (/X VU vi DIFFERENTIAL EQUATIONS {5)6J+af?=2a&*. dx By (6)x2~+y-=ws. (Euler's theorem onHomogeneous Functions.)oxoy Art.43. Art.45. (1)y*=*Ae~ tfte+ti*(2)2-4sinpxsinj?at/. (3)z=>Acoap(ax-y) (4)F=^e~^ar+5y sin^\/(^a+f)>where pand^arepositive. (5)V-cos(pjo?+jp2y-f^2 0). (6)FAe~~risin(m7rx/l)sin(n7ry/l), wheremandnareanyintegers Art.48. ^ (1)-(din x-f^sin3o5+^sin5s+...). 7T (2)2(sina;-^sm2a;-f Jsin3x-...). I*^\-o(^**\-o1 --8m-VT- F-;8ln2a;+(y-w)8lnto- J- o (5)-[|(1+e*)sinx+-|(l-e*)sin /^v32*1 .n-TT/. .W7T W7T\ . (6)^:SssmT\T~n7r<*>*-)*mnx - (7)(a)(2), (3),and(6) ;(6)(6). Miscellaneous Examples onChapter IV. (7)fFenf*sin(nt-gx), where5f--f\/(n/2K). (12)F-~(e-^sina; +^re-9^sin3a; +T^e-26^sin6a (13)Replacea;by ?ra;/i,*by7r2 */P,andthefactorS/TT ANSWERS vii (14)V- g-(e"4*<cos2x+\e~l*Kicos4xH^86*'cos6x+...). 400 (15)F-(e-sinx+\e~9Kisin3x+ie~26*<sin5x+...). 7T [Notice thatalthough F100forallvalues ofxbetween and TT,F=forx=0 orTT,adiscontinuity.] (16)Write 100-Finstead ofFinthesolution of(15). AV (18)F- ^ {e-*"Wcos(7rx/2Z) ^e-9**2^2cos(37TZ/2J) +...}. 7T 4wi (19)^ (sinxcosvf-}sin3xcos3vt+^sin5a?cosM-...). 7T CHAPTER V. Art. 52. (1)(y-2a?-c)(y +3s-c)-0. (2)(2 (3)49(t/-c)2=4x7 .(4)(2 (5)(22/-x2- (6)(y-e"- Art.54. (Thecomplete primitives onlyaregivenhere. Itwillbeseen latei that insome casessingularsolutionsexist.) (1)x (2)x (3)(p- (4)z (5)x=2tan~xp-p"1+c;ylog(p3+p). (6)x (1)x (8)x==sin^p-fo; y**p&inp +cosp. (9)xtanp +c;y**ptanp+logcos p. (10)i-log(jp +l)-log(p-l)-flogjp +c; (11)aj-y/(l+^2)+tan-^; y-c-l/(l-fp2 ). (12) CHAPTER VI. Art. 58. (1)C.P.(y+c)*-x8 ;x~0 isacusp-locus. (2)C.P.(y+c)2-x-2;8.8.x2. viii DIFFERENTIAL EQUATIONS (3)C-P.^ +cy+^-O; S.S.2/2~43a . (4)C.P.y-sin(x +c);S.S.t/2=l. (5)C.P. (2X3+3xy+c)2-4(a;2rfy)* ;x1+1/= isacusp-locus. (6)C.P. c2-I2cxy+8cy*-12afy+16a;3 ;y2-a; isacusp-locus. (7)C.P. c2+Gary-2C?/3-x(3y2- a;)2- ;y2+- isacusp-locus. Art,65. (1)C.P.(</+c)a=z(a;-l)(a;-2);S.S.(-!)( -2)-0;x-l-l/VS isatac-locus andx=*1-f-1/-\/3atac-locus ofimaginary points ofcontact. (2)C.P.(?/+c)2~z(:r-l)2 ;S.S.z=0;a:=1/3isatac-locus; x=l isanode-locus (3)C.P. t/2-2cx-fc2=0; S.S.7/2=x>. (4)C.P.z2+c(z-3?/) +c2=0;S.S.(3y+j)(y-a?)-0 (5)C.P.y-cx2-c2=*0; S.S.x4+4^= ;x==isatac-locus. (6)C.P.y=*c(x-c)2 ;t/=0isaS.S.and alsoaparticular integral; 27y-4o;3:=0is aS.S. (7)Difi.Eq.pycosaa-2pa;yanaa+ya-asmaa-<); S.S.y2cos2a^a;2sin2a;y="0isatac-locus. (8)Diff.Eq.(a?-l)^a-2z?/p-a;2=0; S.S. x=>0 isatac-locus. (9)Diff.Eq.(2x2-fl)p2-f(x2+2x7/ S.S.x2+6xy -f?/2==4;xt/isa tac-locus. (10)Diff.Eq.^(l-z^-Cl-YHO;S.S.-zfclandydbl Art.67. (1)C.P.y-ca+c*;S.S.2-f4y=0. (2)C.P.y-coj+c3 ;S.S.27^2+43-0. (3)C.P.2/=>cz+cose; S.S.(y-xsin-1^)2-!-x1 . (4)C.P.y-c+V(2c2+&2 )JS.S.X2la2+y2 lb2~l. (5)C.P.2/-cx-e;8.8.y=aj(log -l). (6)C.P.y~ex-sin^c;S.S.y=V(a-1)-siirV(l-1/^2 ). (7)J(y-jw?)2a--pia ;2xy=*k2 ,arectangular hyperbola with the axes asasymptotes. (8) (a;-#)a-2&(#-fy)+A;2=>0,aparabola touchingtheaxes. (9)Thefour-cusped hypocycloidx$+y$**k*. Miscellaneous Examples onChapter VI. (1)NoS.S.;z=0 isatac-locus.(2)rP.Y +P/(P-l). (5)2y3a; represent envelopes, y0isbothanenvelope anda cusp-locus. AN8WJECKS (6)C.P.xy~ (7)C.P.x-iyc+xyc*;S.S.y+4z2=0.(Puty-l/F; z-l/X) (8) (i)Putting p+x**3t* weget (ii)C.P.y2+4c21+2cz;S.S.a?-4t/2+4= ;yisatac-locus, (11) C.P. r=a{l-icos($-a)}, afamilyofequalcardioids inscribed in the circle r=2a,which isaS.S. Thepointr= isacusp- locusandaluo aS.S. CHAPTER VII. Art.70. (1)ylogsecx+ax+b. (2) a+t/+51og(y-b). (3)aycos(ax-f6). (4)x=log{sec(ay+b)+tan(ay+1)}+e. (5) 2/ (6)y (7)The circle (x-a)24-(y-&)2=&The differentialequationex- pressesthat theradius ofcurvature isalways equaltok. (9)\/(l+t/j2 )=Jcy2;thecatenary y-b=kcosh{(x-a)/&}. coth(1)y=z(alogx+b). (3)y= (1) (5) (i)Theconicw- (ii)cw=cos.0\/(l (1) (3) (2)y** (4)y= (6) t/== (1) ;y=(a (2)y=|a-logtanArt.73. (2)y-axcos(2logx)+bxsin(2logx). (4) t/=x2 ( Art.74. (2) (3) 4-(1/c-pflf)cos; orcosh0\/(/x/A2~l),accordingas Art. 75. (2)y=.a(*- (4)y-1+6' Art.77. (A)y^( (5) ,v=a Art. 80. cos2x+6sin2a?(5) DIFFERENTIAL EQUATIONS (3)y^{a (4)y-ax+bxrl+(l-arl )e*. (5)y Miscellaneous Examples onChapter VII. (1)y=ae*i*-b. (2) O/j.n+1 gn a:""*1 (4) t/- (5)2/=-ax+61ogx. (6) x 1 (7)y=acosnx+6sinno?+-sinnx^cosnlogsecnx. (8) t/( (9) (i)y-Vtax +fc); (ii)y-V(log*+6). (10)y(acosx+6sinx4-sin2x)e^. (12)y-a**. (14)/--J. (17) (i)y-ae^ +te-^-sinx2 .(Pute=x2 .) (ii)y(l-fx2)-a(l~x2 )+6x.(Putx-tanz.) (18)j-|-2/=2(l-2;2);y=sin2x4-^ cosh (19) 2/= CHAPTER VIII. Art. 83. (1)y=2+x+x2-^x4-^x5 ;exact solutiont/=24-x-fx1 . (2)y=2x-2logx-^(logx)3 ;exact valuet/xH (3) t/ 25=3x24-fx4 (4) t/ (5) t/hasthesame value asinEx. 4. Art. 87. (1)2-19.(2)2-192.(3)(a)4-12,(b)4-118. (4)Errors 0-0018;0-00017;0-000013; Upperlimits 0-0172;0-00286;0-000420. Art.89. M678487; 1-16780250;1-1678449. ANSWERS* XI CHAPTER IX. Art.95. (1)u|l-JI+^-...|.ooBV'*;v= z23s33z43s5 ^44^ 8 8.11 8.11.14 .8(1^___ _ 4.8.12(l+n)(2 +n)(3+n) Togetvfromuchange ninto-n. Ifuismultiplied by theconstant^~^~ f-^theproductiscalled Bessel's function ~- ofordernand isdenoted byJn(x). Art. 96. (1)and(4),allvalues ofx. (2)and(3),|x|<1. Art.97. 22.5 ,2.5.10 (2)w gj-^^ wiscalled Bessel's function oforder zeroand isdenotedby J(x). (3) ,A^ ,!-321.3.5.7 .1.3.5.7.9.11 (4)_x*n._ a-+-_ w-wlogs+2z4 TO DIFFERENTIAL EQUATIONS Art.98. f1 1 1, ()u=*x^~2274 23.4.6X~23T42.6".8* .42.62.8.10x10")' __ 22.42.62 . (2)u=*x (3)u-{l. v-u=ulogx+{- (4)u~{ vwlog+{1~x-5x2-x3+^x4 ...}. Art.99. 0)y (2)^a^l^^a-^i^l^^ 5! [Forsolutions inpowersofl/xseeNo.7oftheMiscellaneous ExamplesattheendofChapter IX.] (3) (4) 2/= Art.100. (1)^+^^4.n-f.Wy-0.(2) (3)y=x2 (1+2a;){a+b\x~*(1+2x)~*e* dx}. where 2I/a;. ANSWERS xiti Miscellaneous Examples onChaster IX. l-f+ll*** *+...}.. 3 9,27+4i"+fi*+io 3 9,27 (2)tt to=(log as)8+2(o-Mlogz)logx 8 6\ CHAPTER XI. Art.113. (1)xla=y/b=z;straightlinesthroughtheorigin. (2)Ix+m^/-f-nz=a;x2+1/24-z2=6;circles. (3) /=az;2-f7/2+22=>62:; circles. (4)x2-y2a;x2-z2=6;theintersections oftwo families ofrect- angular hyperbolic cylinders. (5)x-y~a(z~x)\ (x-y)2(x+y+z)=*b. (6)x2-fy2-fz2=a;y2-2yz-z2*=b; theintersections ofafamilyof spheres with afamilyofrectangular hyperbolic cylinders. (7)yW+n*)- (8)Thehyperboloid y2+z2-2a*-1. (9)(x2-fy2 )(ktan-V)2-*2/2 .(10) l/x-l/y+1/2-1/z+2. Art.114. (1)t/-3s= (2) t/+a (3)xy=*a\ (z2+xy)2-x*=*b.(4) Art.116. (1)cca -ft/24-22=ca ;sphereswiththeoriginascentre. (2)x2+ys+z*=cx;sphereswith centres ontheaxis ofx,passing throughtheorigin. (d)^z-c3 . riv DIFFERENTIAL EQUATIONS (4)yz4-zx+xyc2 ;similar conicoicis withtheoriginascentre. (5)x-cy=ylogz. (6)x2+2yz+2z2c2 ;similar conicoids withtheoriginascentre. Art.117. (1)y^cxlogz. (2)or'y^cze*. (3) (4)y(x+z)=c(y+z). (5) (6)ny-wzc(nx- fo).Thecommon line is Art120. (3)z-ce2-. (4) Miscellaneous Examples onChapter XI. (1)yox;22-x2/-6. (2)y?fz~a\x*+ (3)y+z-ae85 ;yz-z2**b. (4) !/=sinD+C2;/(l+22 ). (5)x2+ir?/2+i*z**t+o. (6) (8)dxlx~dyl2y=dzfiz. (9)y (10) (i)x2+2/2 -i-2:2c(x-f-y+2:); (ii)x*- (iii)yz-yz-xz=*cz*. (14)xy=cezsin^. CHAPTER XII. Art.123. (1) </>(z/z,y/)=0. (2) (fa4-my4-wz,x2+1/24-2 )0. (3)^{y/, (x24-y2+^)/}=0.(4) (z2- 1/2 ,x2-22 )-0. (5)<{(z-y)2(z-fy+2),(x-y)/(2-o;)}0. (6)0{z2+y2+z2 ,y2-2y2-z2 }=0. (7) (8) (9)y2=4xz. (10)a(z2- (12) (f>(xz+y2 ,z) ;surfaces ofrevolution about theaxis ofz. Art.126. (1)$(Z+X19X!+X2,X1+X3)^0. (2)0(z, (3) (4) (5)0(4V^~^32 >2x3~x22 ,2x2-x12)=0; special integral z=0. (6)^{ap-Sajp~3x9f+6V(3~i>-3g-a?)}0; special integral ANSWERS xv Art.129. (1)z(2&a+Da;+01/4-0. (^)=a;cosa-f /sina-fc. (3)z=ax-ft/loga*fc. (4)2a3x+a~2 t/-fc. (5)2=2zseca +2ytana +c.(6)2=x(l+a)+y(l+l/a)H-c. Art.130. (1)az**(x +at/+6)2 , (2)2=dbcosh{(z +a (3)22-a2~(z+a*/+6)2 ,or2=6.(4)22(1 (5)(2+a)e*+y~&. (6)2 Art.131. (1)32=2(z+a)*+3a*/+3&.(2)2a2 (3)az-ax2-fa2a;+ea^4-a6. (4)(22 (5)z=*a(ex+ey)+b. (6)az=a2x-fasinx+siny4-a6. Art.133. (1)--2-loga;y. (2)3*=xy-x2- 1/2 .(3)83--27z2 ?/. (4)zx**-y. (5)2=0. (6)z2-!.(7)2-0. Art.136. (1)4s~-y2 . (4)Aparticularcase ofthegeneral integral, representingthesurface generated bycharacteristicspassing through thepoint (0,-1,0). Miscellaneous Examples onChapter XII. (1)zax-fby-a?b;singular integralz2 x*y. (2)zx=ax+by-a?b;singular integral22=y. (3)0{^,(z2+^)2-a;4}0. (4)z3X3-3az2-fa?x+2i/4-4a^+3a2y2-a*y+6. (5)z=*axl+61oga; 2+(a2+26)ic 8-1-fc. (6)*-#{(! +*a)/i.*ii- 8*}- (7)3a(x-fay+6)=(1+a8 )log2,orz6.= isincluded in2=6,but itisalsoasingular integral. (8)z(l+a^ +Vt)^(x 1+ax2+bxB+c)^. (9)^(s-S-e4 *!,ze***, 2-J-e^)=.0. (10)2aa;-( (11)saax2-(2+3a+JasV+&. (12)22*(l (13)zatan(x+ay+6),or2=6.2= isasingular integral, but itis alsoincluded in2=6. (14)s2ax2+6y2-3a8+b*.Singular integral2-2a?/9-y*/4. (15)*-3+y~l2V{(*-lHy-l)}. (16)2-xy-a xvl DIFFERENTIAL EQUATIONS (17) 0(z/x, zly)=Q;cones with theoriginasvertey. (18)a?24-y24-za-'2xcosa4-2ysina4-c;spheres with centres onthe given circle. Thegeneral integral gives other solutions. (19)xyz^c. (Thisisthesingular integral. Thecomplete integral givesthetangent planes.) (20)The differentialequation (z-px-qy)(l-llp~llq)-*Qhasno singular integral, andthecomplete integral represents planea. Every integralincluded inthegeneral integral represents the envelopeofaplane whoseequationcontainsonly one parameter,thatis,adevelopable surface. CHAPTER XIII. Art.139. (1)ya{(x-a)24Y+2z}~&. (2) (3)z=ax+bev(y+a)-*. (4)22=2( (5)z~ax+3a?y+b.(6)(z2-fa (7)z-x*+ax+l(y +a)*'*+b. (8)z=ax Art.14L (1)2;a1 (2)z**alxl+afl 2.sin"1(a1a2^3)-fa8. (3)2axlogxl+a2logx2 3V(ai+ (4)22=a^!2+a2#224-a3a?32-2(a (5)2(a 1a2a3)l/8logz=&\x>\+2a?22 "*"asx32+! (6)4a^=-4ax2logjr3+2a1a2(x1-o;2)-(xj+x (7)(14-a^g) log2(aa+a2)(xt4-a^g4-a2T34-a3). (8)2-(a 14-o2)o;14-(2a 1-a2)a;2-t-(~a Art.142. (1)zdt(a?i4-a?2)24-log#3-ha. (2)Nocommon integral. (3)z=* a?!24-#224-x324-a,orz=Xj24-2x2x8-fa. (4)za(xx4-2x2)4-6logx34-2a6logx44-c. (5)2=a(3x 14-x23~x33)4-6. (6)Nocommonintegral. (7)i?a(x1~x4)4-6(x 2-x3)4-c, orZ=a(x 1-2x2)4-6(2x 3-x4)4- (8)^-^(S^ +V-^a8 )- (9)-0(x 1-x4,Xa-^s)*or2=0(x 1~2x2,2x3-x4). Miscellaneous Examples onChapter XIII. (1)2*-axlogx1-aja2logx24-a2logxs4-a3. (2)Nocommonintegral. (3) aj ANSWERS xvii (4)-&!log XJL (5)21ogz~crfc(x 12+x2*+ay8 ). (6)afl- (7)4z+x^-fz22+x32-0.(10)2= (11) (iii)Sz^x^-SXiXj+c. CHAPTER XIV. Art.144. (1)*-x*+x/(</)+*X</). (2) (3)--1sinxy+yf(x)+F(x). (4)-ay+/(y) logcc+ (5)^sin(a;+2/)4-i/(a;)4-J?t (2/). (6)*-- (7)2(2 -ft/2 )2-l.(8) (9)2(x2+i/2 )a .(10) Art.145. (1)2:=J?'1(y+a;)+F2(t/+2x)+^3(i/-f3aJ). (2)-/(y-2x) +J(2y-x). (3)*-f(y +x)+F(y-x). (4)The coiiicoid 4X2-8xy+ya+8z- Art.146. (1)2=/(2y-3x)+xl'(2y-3x). (2) (3)(2=-/(2/ +2x)+x^T (2/+2x)+^(2/). (4) Art.147. (1)3=x4+2x3 ?/-f/(t/-f-x)-fxJF(2/+x). (2)z-6x2 t/+Sx3+/(y+2x)-fF(2y-fx). (3)F-- Art.148. (1)z- (2)2= (3)2--x2eos(2x +y)+/(y +2x (4)2 (5) (6)2 Art.149. (1)z**X8iny+f(y-x) +xF(y--x). (2)x*+2xs y-f-/(f/4-5x) +J?(y-3x). (3)z=sinx-ycosx+f(y-3x)+F(y+2x1, xviii DIFFERENTIAL EQUATIONS (5)z-* (6)yxlogt4-1loga?+f(t4-2z)4-JF(*-2x). Art.150. (1)*-/(*) +^(y)+68Xt/+2x). (2)*--{/(y-)4-*J(y-)}. (3)7- (4)*-/(y+x)4e~*F(y- a?). (5)z- (6)F^S^^-H"*^ (7)*- (8)c-l+-{(y-*)-l). Art.151. (1)2=- (2)ii-l+aj-y-xy +^/^+^ -3^/)+9cos(a- l (+*). (5)y (6)2-6"{otan(y+3)+*/(y+3a;)+^(y Art.152. (1)y*r-2ys+<p+6y. (2) jrf-g*- (3) (4) (5)2pr+^-2^(ri-a)l. (6) Art.154. (1)-/(y+sina?) f^(y-sina?). (2) (3)y~^(x +y+z)~<f>(x),or/ (4) /(a?+tany)+JF(a?-tan y). (5) (6)y-/(*-f +*)+a?J?'( +y+). (7) Art.157. (1)p+3-2y/(g-23 +3y);X--^ (2)p-*-/(j-y); Xoo. (3)y---/(-2y) (4)p-y=f(q +x);p+y-F(q-x); X-dbl. (5)p-y-/(j-2x); p-2y~F(q-x)i X-lor -J. (6)px-y~f(qy-x)\ X~xor -y. (7)sp-x/(2g-y) ;X- Art.158. (1)*aa;-f-&y-Jir2+2a;y-|ya-H(?; 25Ja;2(l-f3m2 )-f(24-3mjay-fn4-0(y4-wi) -J(x24-3y2 )4-n*4-^(y+ww)- ANSWERS (2)z (3)z (4)* (5)x (6)2;4-t//m+wx-wlogx<J>{xm y) ;theothermethod fails. (7)22=x2+2/2-f2ax4-26t/+c; ^x^+i/2*2nx+i/r(y+wx). (8)2z= t/2-x2 . Miscellaneous Example onChapter XIV. (1)-*y+*/(y)+*(y). (2) (3)yz (4)z (5) /(*/+logx)+x-F(t/+logx). (6) (7)* (8)4z=6x2/-3x2- 42=6x?/-3s2- (9)3^-3c2(a;-h (10)mz-fsiny-fm2sinxmnx=m^>(y+mx). (11)2 (12)2=^+2/3+(a;-ff/-f-l)a . (13) z (20)pa+^y~f(p*+q*)\py-qx F(q/p). Miscellaneous Examples ontheWhole Book. U)(a?-t/2 )2=c*t/. (2)y=x*+ce~*\ (3)2secx8ec2/=ix4-sinxcosaj +c.(4)(xy+c)z=(x2+y)(y2~cx). (5)1+xy^tKc +sin^aOv^l-x2 ). (6) 2/=(^- Jo?)cos2x+J5sin2ac <J5/OQ1 (7)y-.~^~++~xea (sin2x-cos2a?)-fAe~* -f^e35cos(2a? 4-a),D^01ZD lo (8)yJ4-Bx+Cxlogx-flogx-fjx (log )*+Ja^. (9) t/+sec#=ctan x. (10)aj-4e2/+5e-2'-|(co8-Bin); y=^e2t-35e~2<-|cos *. (11)x2/3(y-l)2/s-fc;S.S. ?/-1. (12)y=acosec(6-x). / X2\ / Xs\ (13)y-(^+-Bx+Jsin2x+(E+Fx-- Jcos2x. (14)2x2/=3x2-fc. (15)z+xy=*c(x +y-xy). (16)a^+y>+i-ca^. (17)z~f(xy)-$x*-$y*. (18) (19) (20)2-ax-fby+a2 -f-62 ;singular integral4*+x2+y20. (21)*-/(*- XX DIFFERENTIAL EQUATIONS (22)z**ax*+by+4af ;singular integral162-1-z*0. (23)s./fc +yJ+Jte-^^ +y8 ). (24)z-xf(y)+yF(x). (25) cs-(s+a)(y+6). (26)z=lxy+f(y/x)+xF(ylx). (27)a-/(*+)+JP (28)y(a;+c)=&x;singularsolutionsy=andy+4z2-0._ (31)z2+y2+322(a;cosa +y8ma+c). (32)y-e'-fe8"+$*". (33)x=e-*'(acos\J+6sinX<)4-cos(^/-a), where -4/V{(^2+X2-^2 )2+4*VKtana-2^/(/c2+X8- andaand6arearbitrary constants. (34)y^Acos (sinx)+Bsin(sin x). (35) (i)^-^log(r4-^) +5; (ii)<f>-A\ff-W*df+B; (36)F=^{Hf where r2x2+y2+22 . (39)u-o(l-f- x x\a (41)y-a= (42)y=(l If2a isaninteger,theintegral canbeevaluated by putting z=*(l +#)/(! -x). (43) (i)y=(l-x*)(A +Blogx)', (ii)y-(l-a-)(aj +4+Blog*). (44) (1-x2 )y=(a+6 Je-*2 te)c**2 -[Put logy- J(u-JP)rf. t4-xis asolution ofthedifferential equationinu.] l (2w-l)2! (2n-i)(2n-2)(2n-3)i! (46)y-^Iz5+By?+E(x2+1),replacing C/6byB. 6! BJ (J 3!Vo' bothconvergewithin thecircle\x\= |a|. ANSWERS XX) (60) (51 )ir/5p r5o^ p>i5 =- !must bsafunction ofxalone;x*v- Q\dy dx)y (52)we"-avVdx+6,where v-#/Pandw 1vdx. (53) (54)y(1- a?)-4(3-2x)e2*+J5(l-2x)e~2a> . (56) xS+ys-cfy +z). (57)y-4e-a*-fex (Bcosx\/3+sinx\/3) Yirhrve~2x {157x(6 cosx+11sinx} +3(783cosx-56sinx)}. (58) (59) //?ov /-xT>. (62) (i)rutV" 1/l/ y(59)24(x+t/)4(x2+t/2H-02)-c(a;2-fy2-2 ). (60)x* ^W/'-x1x(c+tanx) -~~ --r~7~ -^^-- x)dx'v/*21-ctanx [SeeEx.41formethod.] (65)Ifaparticle Pmoves sothat itsvelocityisproportionaltoth radius vectorOPand isperpendiculartoOPand alsotoa fixed lineOK,then itwilldescribe with constant speed a circle ofwhichOK istheaxis. (67)f2sin2(#+a)-l ;singularsolution r4-!. (68) t/2-x2=ex+2a* a\/(4a2-c1 );singularsolution y1x2=2ay. (70)4a(y-c)-(x-c)a ;singularsolutiony-x-a. (71)x+accos^4-clogtan J0. (72)acos +&cos0'-i. (74) 2cf/=(x-t-c)2 ;singularsolution y(y-2x)0. (75)x-f^y+ay20;(y+apJv^-t-lJ-c-f asinh"^, %V(P2+l)+p(o +asinh""1^)=0. There isnosingularsolution. Thep-discriminant y2-4ax represents thecusp-locusoftheinvolutes. (77)y~ax, s-fc+v^+S1 );*-V(^ +y*)+/(/*) Thesubsidiary integrals representafamilyofplanes through theaxis ofzandafamilyofrightcircular cones with theaxis ofzasaxis;thegeneral integral representsafamilyofsurfaces each ofwhich contains aninfinite number ofthepairsofstraight lines inwhich theplanes andcones intersect. (78) (79) (80) (ax-by)t(z+c)-/{(ax +fy)/(s- c)}, DIFFERENTIAL EQUATIONS (81) (i)I~EIR +Ae-RttL ;(ii)A^I^E/R; (iii)I-E/R. (82)7-acos(pt-e)+Ae-RtlL ,where a-/yXIP+Z*p),tane andAisarbitrary. (83)Q-asin(jrf-e),where tane-(CLp1-l)/pCJ? and a-SCW{(OLp-1)2+p2C2^2 }. (85)x~Acos(t-a) +BcoB(3t-/3); y~'2Acos(<-a)-55 cos(3*-) (86)aand6aretheroots of\*(LN-M*)+\(RN +LS)+RS~Q. (91)x**A cos(pt-a) +cos(qt~/3), y^Asin(^-a)-Bsin (JJ-/3) where2^p-V(4c2+^)+*2S-\/(4c2+JK1 )-<c. (92)J-f(a+6)^+a6^a6c. (93)py^(n2~2/x2 )makes tneamplitudeoftheparticular integrala maximum, provided 2ju?doesnotexceed n2 . (94)x4e~Hcos(^-6),where^p\/(n2~^2 )- (97)0=*FaV-acos0. (98)ysin(p6/c)-Asin(ps/c)cos(pt+a) (100)0Ccosh m(y-f A)cos(mx~n<). (115) (vi) a=4(-2 (viii)MZ=2*P cos-~+gsin y**-xv. (119)u=cos sinmt.* 'm c (120)z Noteonalternative forms ofanswers. Inseveralexamplesaslightvariation inthemethod ofsolution may leadtoadifferent form ofthecomplete primitive. Thus inEx. 3,Art. 70,theanswergivenisaiy=cos(ax+6),butthestudent may equally wellobtainay=sin(ax+6),oray^sinh (ax+b).Ifinthe firstform bisreplaced by(b-|?r)weobtain thesecond, while ifinthesecond a and6arereplaced byaiand birespectively, weobtain thethird after division byt.Other formsmaybeobtained byreplacingabyI/a. Intheanswer toEx. 4,Art. 116, c2jnaybereplaced by-c2 ,orc, or c.Ingeneral anarbitrary constant must besupposedtohave allvalues, real, imaginary,orcomplex, andmaybereplaced byany function ofanewarbitraryconstant. Wherepairsofintegralsareneeded, alternative pairsoften arisevery naturally. Thus theanswers toExs.5and 6,Art. 113,maybereplaced fcy andby ANSWERS respectively.Inthis setofexamplesthepairsu=a,t?6may be replaced byf(u,v)a, F(u, v)=6,where/ andFareanytwo inde- pendent functions ofuand v. Alternative answers aretobefound forseveral oftheexample on partialdifferentialequations, e.g. sina=^-cosaforEx.3,Art. 42,oxoy and 22(a~y2 )= (+6)2forEx. 2,Art.139(seenoteonp.171). INDEX (Thenumbersrefertothepages.) Adams, 224. Adams' numerical method, 224. Adjoint equations, 255. Ampere, xvi, 183. Angstrftm's determination ofdiffusivity, 58. Apparent singularity, 213. Approximate methods, 5,94,224,247. Arbitrary constants, 2,60,126, 127, 252. Arbitrary functions, 49,137, 147, 172. Asymptotic series, 217,251. Auxiliary equation, xv,26,174,254. Bar, vibrating, 190. Bateraan, 222,232. Bernoulli, xv,12,18. Bernoulli's equation, 18. Bessel, 110. Bessel's equation, 114, 116, 118, 120, 214, 217, 253. Boole, xv. Boundaries, discriminant-loci as,195. Boundary conditions, 53,66. Briot andBouquet,xvi. Brodetsky's graphical method, vi,6. Bromwich, 247. Cauchy, xvi, 121, 124. Cayley,xv. c-diseriminant, 67,155. Changeofvariables, 40,61,79,85,91, 93,119, 120, 164. Characteristic index, 214. Characteristics, 6,97,158. Charpit, xvi, 162. Charpit's method, 162. Chemistry, 244. Chrystal, xvi, 150. Clairaut, xv,76. Clairaut's form, 76,79,195, 196, 199. Commonprimitive,10. Complementary function, 29,87,175, 254. Complete integral, 153. Complete primitive,4.Conditions ofintegrability, 139, 144, 229, 231. Conduction ofheat, 52,53,57,58,59, 60,250. Coniluent hypergeometric equation, 218. Confocal conies, 23,79. Conjugate functions, 24,189. Constant coefficients, xv,25,49,173, 178,250, 252, 254. Constants, arbitrary, 2,50,126, 127, 252. Convergence, xvi,112, 124. Corpuscle, pathofa,48. Cross-ratio, 202. Cusp-locus, 68,73,195, 198. D'AIembert, xv,25,44,49. Darboux, xvi. DefiniteIntegrals, solutionby,250,25L Degree,2. Depressionoforder, 81. Developable surface, 189. Difference equations, 254. Difficulties, special,ofpartialdifferen- tialequations,51. Diffusion ofsalt, 60. Discriminant, 67,71,155, 194. Duality, 160, 161, 189,248. Dynamics, 2,24,28,36,46,47,50,61, 85,86,190,242-249. Earth, age of,60,212. Einstein, 247. Electricity, 24,29,46,48,58,59,134, 241-244. Elimination, 2,49,50,179, 194. Envelope, 66,71,146,155,192,195,196, 200. Equivalence,92. Euler, xv,12,25,49. Exact equations, 12,23,91,191. Existence theorems, 121,252. Factorisation oftheoperator, 86. Falling body, 24,86. Falling chain, 246. XXVI INDEX (The numbersrefertothepages.) Finite differences, 253, 254. First order and first degree, ordinary, 12,133;partial, 147, 151. First order buthigher degree, ordinary, 62,65;partial, 153,162, 165. Fontaine, xv. Forsyth, 150,232. Foucault's pendulum, 247. Fourier, 54. Fourier'sintegral,60. Fourier's series, 54. Frobenius, xvi, 109. Frobenius' method, 109,127,208. Fuchs, xvi. Fuchsian type, equations of,213, 214. Fuchs' theorem, 211. Functions, arbitrary, 49,137, 147, 172. Gauss, 110. General integral, xvi,137,147,149,157. General solution, 4. Geometry, 5,19,65,133, 137, 146, 173, 188, 189, 192,255. Goursat, xvi,172,232. Graphical methods, 5,8. Groups, xvi,120,232. Hamilton's equations, 248. Heat, 52,53,57,58,59,60,250. Heaviside, 58,61. Heun, 94. Heun's numerical method, 104. Hill,M.J.M.,vi,xv,xvi, 65,150,155, 196,230,232. Homogeneous equations, xv,14,40,44, 83,144, 171, 173,205,251. Homogeneouslinear equations, 40,44, 171,173,251. Hydrodynamics,246. Hypergeometric equation, 119,120,214. Hypergeometric series, 92,119. Indicial equation, 109, 111. Inflexion, locus ofpoints of,200. Initial conditions, 4,28,53. Inspection, integration by,12,172. Integrating factor, xv,13,17,22,23,91, 205, 237,265. Integrability, 139, 144,229,231. Integral equation, 96. Intermediateintegral, 181. Invariant, 92. Jacobi, xvi,165. Jacobi's Last Multiplier, 249. Jacobi's method, 165,231,248.Kelvin, 58,60,250. Klein, xvi. Kutta, 94,104, 108. Kutta's numerical method, 104. Lagrange, xv,49,81,162. Lagrange's dynamical equations, 248. Lagrange's equation, 254. Lagrange's Hnearpartial differential equation, xvi,147,151, 158,230. Laplace,xvi. Laplace's equation, 61,189, 190, 234, 235,261. Lastmultiplier, 249. Laws ofalgebra, 30. Legendre,110. Legendro's equation, 117, 120,214. Leibniz, xv. Lie, v,xvi,232. Linear differenceequations,254. Linear equations (ordinary), ofthe first order, 16,252;ofthesecond order, 86,87,88,109, 127,208,252; withconstant coefficients, xv,25,252. Linear equations (partial),ofthe first order, xvi, 60,147, 151, 158,192; with constant coefficients, 49,173, 178,250. Linearly independent integrals, 253. Lines offorce, 24,134. Liouville's solution ofthewave equa- tion, 220. Lobatto, xv. Maxwell's equations, 69. Mayer's method, 206. Mechanics, seeDynamic*. Membrane, vibrating, 190. Monge, xvi, 172. Monge's method, 181,183. Multipliers, 135,248,249. Newton, xv. Node-locus, 68,196. Non-integrable equations, 142. Normal form, 91,92. Normal integrals, 215. Normal modes ofvibration, 242,244. Number oflinearly independentinte- grals,253. Numerical approximation, 94,224. One integral used tofindanother 87, 136. Operator />,30,44,86,174, 252, Operator 0,44. Orbits, planetary, 86,247. DIFFERENTIAL EQUATIONS xxvii (Thenumbersrefertothepages.) Order, 2. Ordinary point,212. Orthogonal trajectories, xv,20,23,138, 189. Oscillations, xv, 2,28,29,36,46,47, 48,50,61,190,241-245. Page, 232. Particular integral, xv,4,29,33,44,87, 175, 178, 195,254. p-discriminant, 71,155. Pendulum, 28,244,245, 247. Perihelion ofMercury, 247. Physics,seeConduction ofheat, Cor- puscle tDiffusion, Dynamics,Electri- city, Hydrodynamics, Potential, Ra- dium, Resonance, Telephone, Vapori- sation, Vibrations, Wave equation,etc. Picard, xvi, 94,121. Picard's method, xvi,94,122. Poincar6, xvi. Poisson's bracket expression (F,F^),166. Poisson's method, 189. Poisson's solution oftheWave equation, 220. Potential, 134, 190. Power series, xv,xvi, 4,109, 124. Primitive, 4. Radium, 24. Realsingularity, 213. Reduction oforder, 81. Regular integrals, 110, 118,208. Regular singular point,212. Remes' numerical method, 227. Resonance, 37,46,243. Riccati, 110. Riccati's equation, 201. Riemann, vi,232. Riemann's P-equation,214. Runge, xvi, 94,99,100. Runge's numerical method, 99. Schwarz, xvi, 92. Schwarzian derivative, 92. Schlesinger, 232. Sohrodinger's equation,222. Second integral found byusing afirst, 87,136. Separation ofthevariables, xv,13. Series, solution in,xv,xvi, 4,109, 124. Shaft, rotating,47. Simple harmonic motion, 2,85,242,244.Simultaneous equations, 42,59, 133, 168,171,252. Singular integral,155. Singular point, 7,212. Singular solution, xv,4,65,192. Solid geometry, 133, 137, 146, 173, 188, 189. Solvingforp,x,ory,62. Special integral, 137, 150,230. Standard forms, 153. String, vibrating, xv,50,61,190, 218, 246. Subnormalintegrals, 215. Subsidiary equations, 147, 164, 166. Substitutions, 40,61,79,85,91,93,119, 120, 164. Sylvester's dialytic method ofelimina- tion, 194. Symbolical methods, xv,33,44,46,61, * 175, 178,252. Tac-locus, 72,195. Taylor, xv. Telephone, 58. Todd, 213. Total differential equations, 137,205. Transformations, 40,61,79,85,91,93, 119, 120, 164. Transformer, electrical, 48. Vaporisation, 24. Variation ofparameters, 88,93. Vibrating strings, equation of,50,61, 218,256. Vibrations, xv,2,28,29,36,46,47,48, 60,61,190,218,241-215. Wada, xvi, 5,8,9. Wave equation, 219. Wave mechanics, 222. Weber, 194. Whittaker andWatson, 252. Whittaker's solution ofLaplace's equa- tion, 51,251. Whittaker's solution oftheWave equa- tion, 222. Wronski, 253. Wronskian, 253. xabsent, 82. yabsent, 82. Zeeman effect, 244.