Solving a differential equation of the form $y''(x)+(b+ax^2)y(x)=0$ I don't know any method to solve an equation of the type: 
$$y''(x)+(b+ax^2)y(x)=0$$
Me asking this questions is because I need to understand why solving this in quantum mechanics for a harmonic oscillator gives:
$$\frac{-\hbar^2}{2m}\frac{{\rm d^2}\psi(x)}{{\rm d}x^2}+\frac12\underbrace{(m\omega^2)}_{k}x^2\psi(x)={\rm E}\psi(x)\\\longrightarrow\psi_n(x)=\sqrt{\frac1{2^nn!}\sqrt{\frac{m\omega}{\pi\hbar}}}\exp\left(-\frac{m\omega x^2}{2\hbar}\right){\rm H}_n\left(x\sqrt{\frac{m\omega}{\hbar}}\right)$$
where ${\rm H}_n$ are the hermite polynomials:
$${\rm H}_n(x)=(-1)^ne^{x^2}\frac{{\rm d}^n}{{\rm d}x}\left(e^{-x^2}\right)$$
Sorry I don't know how to solve these equations. I can solve atmost:
$$y''(x)+ax=b$$
 A: This is a Sturm-Liouville problem
$$
L y = \left[(d/dx)^2 + a x^2 \right] y = (-b)\, y
$$
see here. And the $H_n$ are the eigenfunctions for that problem or closely related to them.
$$
Ly = \lambda y
$$
is an eigenvalue problem, now for vectors $y$ of a Hilbert space.
The spectrum seems discrete, that is why $n$ shows up in the solutions. This happens e.g. if there are boundary conditions which restrict the $y$. 
Here the $H_n$ are given by a compact definition, which results in polynomials of order $n$ or related to $n$.
If they are not given this way, one would probably assuming some polynomial $P_n$, applying $L$ to it and matching the coefficients to fulfill $L P_n = - b P_n$. Usually some boundary conditions are needed as well to make the problem unique enough. Starting with a low degree one and then using some sort of Gram-Schmidt to get the resulting solutions normalized and orthogonal.
A: In general you can't solve that differential equation without special functions.  For example, you can express the general solution in terms of Whittaker M and W functions.  
