You know, I just realized there is no forum section for differential equations...

Anyway, I came across an ODE that apparently has no closed form solution, yet does not seem to fall into the category of any of the major forms of series solutions that I know of (Bessel, Hankel, Hermite, etc.)

$\displaystyle y''+(-ax+b)y=0$

I figured it was probably it was a version of the Confluent Hypergeometric equation (which practically every series solution in Physics seems to satisfy), but I have yet to find a transformation to make it into one. (Obviously I can expand the solution in terms of, say, Bessel functions but that seems like cheating to me and doesn't really address what the solutions might represent.)

So I was wondering if anyone recognizes it, can transform it into something more recognizable, or could tell me of a website, book, etc. that could tell me what it might be. (I refuse to believe it is unexplored...I got it from the Schrodinger equation for a linear potential, which corresponds to a simple gravity problem.)

Thanks!

-Dan