# Differential Equation

• Mar 28th 2006, 08:37 AM
topsquark
Differential Equation
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
• Mar 28th 2006, 08:41 AM
TD!
According to Mathematica, it can be solved in terms of the Airy-function.
I have no knowledge of those, but you could take a look at Mathworld's page on the Airy Functions.
• Mar 28th 2006, 10:20 AM
topsquark
Quote:

Originally Posted by TD!
According to Mathematica, it can be solved in terms of the Airy-function.
I have no knowledge of those, but you could take a look at Mathworld's page on the Airy Functions.

Well that would explain it. Airy functions aren't in my Mathematical Physics book. I have heard of them, though. Thanks for the tip!

-Dan