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Homework Help: Trouble with a 2nd Order D.E.

  1. Oct 10, 2009 #1
    1. The problem statement, all variables and given/known data

    [tex] \frac{d^2\phi(\eta)}{d\eta^2} = (\eta^2 - K) \phi(\eta) [/tex]

    Where K is essentially a constant, K = 2n + 1 (n is an integer).

    3. The attempt at a solution

    I don't even know where to begin since [tex]\phi[/tex] is a function of [tex]\eta[/tex]. A push in the right direction would be much appreciated.
  2. jcsd
  3. Oct 10, 2009 #2


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    Stuff that looks like that usually wind up having solutions that are special functions (like Bessel etc). That appears to be a parabolic cylinder function. I got that by creative googling. It may have a simpler form for the case K = 2n + 1. Don't know. But that will give you a start for researching it. What kind of course is this? Are you supposed to be able solve it simply?
  4. Oct 11, 2009 #3
    Thanks for the reply. This is for my quantum mechanics course, and the equation I set up relates to solving the time-independent Schrodinger equation for the harmonic oscillator in momentum-space. My textbook solved a similar DE using Hermite polynomials, but I was hoping there was a simpler solution. I'll search for special functions & "parabolic cylinder function." Thanks again for the reply.
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