From: israel@math.ubc.ca (Robert Israel)
Subject: Re: analytic solution to differential equation
Date: 27 Apr 1999 16:16:16 GMT
Newsgroups: sci.math,sci.math.num-analysis,sci.math
Keywords: Whittaker functions

In article <372527E8.A9BEB39E@hotmail.com>,
Desipoet  <desipoet@hotmail.com> wrote:

>I need to solve analytically the following differential equation:
>
>d2y/dx2 (second derivative of y wrt x) = k (A+B/x) y
>
>where k is a constant.

According to Maple:

y(x) = _C1*WhittakerW(-1/2*k*B/(sqrt(k*A)),1/2,2*sqrt(k*A)*x)
+_C2*WhittakerM(-1/2*k*B/(sqrt(k*A)),1/2,2*sqrt(k*A)*x)

where _C1 and _C2 are arbitrary constants, and WhittakerW and 
WhittakerM are special functions (see Abramowitz and Stegun,
Handbook of Mathematical Functions).  Actually it seems these
functions were defined in order to solve exactly this type of
differential equation: according to Maple's help page
-------------
The Whittaker functions WhittakerM(mu, nu, z) and WhittakerW(mu, nu, z) 
solve the differential equation 

                           /                       2\
                           |         mu    1/4 - nu |
                     y'' + |- 1/4 + ---- + ---------| y = 0
                           |          z         2   |
                           \                   z    /
-------------

Robert Israel                                israel@math.ubc.ca
Department of Mathematics        http://www.math.ubc.ca/~israel 
University of British Columbia            
Vancouver, BC, Canada V6T 1Z2