Can the Parabolic Cylinder Function Solve a 2nd Order D.E. with a Constant?

[hotcommodity]

Homework Statement

$$\frac{d^2\phi(\eta)}{d\eta^2} = (\eta^2 - K) \phi(\eta)$$

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

The Attempt at a Solution

I don't even know where to begin since $$\phi$$ is a function of $$\eta$$. A push in the right direction would be much appreciated.

 

[Dick]

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?

 

[hotcommodity]

Thanks for the reply. This is for my quantum mechanics course, and the equation I set up relates to solving the time-independent Schrödinger 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.