Laplaces ODE Method
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A brief note by Phil dated 12.5.08 on Laplace's Method, which solves ODEs whose coefficients are at most linear in the variable by writing solutions as contour integrals. It includes Google Books excerpts from Brian Davies' Integral Transforms and their Applications. Phil comments on the link to Laplace transforms, to generating functions (Hermite example), and to the confluent hypergeometric W1 and W2 functions in Coulomb scattering (Messiah).
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Laplace's Method for solving ODEs PhL 12.5.08
The subject is a method for finding solutions of ODE's which are integral representations with somewhat adjustable integration paths. The method works when the ODE coefficients are at worst linear functions of the variable, which is a quite common situation. The method is called Laplace's Method. Laplace did so much stuff that it is hard to extract this specific subject on the web, which is different from Laplace Transforms and other methods he had for doing things. This subject might be a standard one in ODE books of which I have none.
These are some Google book pages taken from
http://books.google.com/books?id=FViAkJi9c7EC
Integral Transforms and their Applications by Brian Davies (Hardcover - Jan 2, 2002) $56 Amazon
Here is the full link that got to these pages directly:
http://books.google.com/books?id=FViAkJi9c7EC&pg=PA303&lpg=PA303&dq=%22laplace's+method%22+%22integral+representation%22&source=web&ots=IXwWCKdvej&sig=PAk8wHI7WTpt8tdkXExvBkewnJY&hl=en&sa=X&oi=book_result&resnum=4&ct=result#PPA324,M1
I think this little snippet teaches several things of interest to me. One is how it is just an extension of the Laplace Transform idea to a different set of contours. Another is that it sheds light on the generating function idea, ad least for Hermites as in his example. The next page after my last was not on line. This subject relates directly to the W1 and W2 functions which are confluent hypergeometric functions involved in Coulomb scattering, see for example Messiah Vol 1 page 481.