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Math Help - Differential Equation

  1. #1
    Forum Admin topsquark's Avatar
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    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.)

    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
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  2. #2
    TD!
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    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.
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  3. #3
    Forum Admin topsquark's Avatar
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    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
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