Solving 2nd Order Diff Eq in Quantum Mechanics

[ELog]

So I was trying a few quantum mechanics problems and encountered this diff eq:

$$\frac{\hbar^2}{2m}\frac{\partial^2}{\partialx^2}\psi(x) + \frac{1}{2}kx^2\psi(x) = E\psi(x)$$

I put it into the form:

$$\frac{\partial^2}{\partialx^2}\psi(x) + (\frac{2mE}{\hbar^2} - \frac{m}{\hbar^2}kx^2)\psi(x) = 0$$

But there is where I'm having difficulty. Is there an alternative form that would be easier to solve, or is there any way to reduce the order of the diff eq? I'm not used to second-orders with variable coefficients. Any guidance is greatly appreciated.

NOTE: k is an (apparently arbitrary) constant and has no quantum meaning in this context.

 

[coelho]

The solutions of this differential equation are related to the Hermite polynomials. Look at:

http://en.wikipedia.org/wiki/Hermite_polynomials#Hermite_functions