Solving a second order linear differential equation Am having difficulty obtaining a solution for this ODE. 
$ (\omega^2-x^{\alpha})\psi^{''} - \alpha x^{\alpha-1}\psi^{'} - \gamma \psi = 0 $
where $ \omega $ and $ \gamma $ are positive constants, $ x $ is the independent variable and $ \psi $ is a function of $ x $. I need the solution for this ODE for general $ \alpha $. I tried using the series method of solution to solve (frobenius) but I wasn't successful.
 A: It's not pretty, but here's what I found when $\alpha$ is a positive integer.
The first step I took was using product rule, a fairly standard procedure in ODEs.
$$(\omega^2-x^{\alpha})\psi^{''} - \alpha x^{\alpha-1}\psi^{'} - \gamma \psi = 0$$
$$((\omega^2-x^{\alpha})\psi^{'})' - \gamma \psi = 0$$
$$((\omega^2-x^{\alpha})\psi^{'})'=\gamma \psi$$
Now applying the series method we set $\psi=\sum_{n=0}^{\infty}a_nx^n$, and $a_n=0$ if $n$ is negative
$$((\omega^2-x^{\alpha})\sum_{n=1}^{\infty}na_nx^{n-1})'=\sum_{n=0}^{\infty}a_nx^n$$
$$\bigg(\sum_{n=1}^{\infty}\omega^2na_nx^{n-1}-\sum_{n=1}^{\infty}na_nx^{\alpha+n-1}\bigg)'=\sum_{n=0}^{\infty}a_nx^n$$
$$\sum_{n=2}^{\infty}\omega^2(n)(n-1)a_{n}x^{n-2}-\sum_{n=2}^{\infty}n(\alpha+n-1)a_nx^{n+\alpha-2}=\sum_{n=0}^{\infty}a_nx^n$$
$$\sum_{n=0}^{\infty}\omega^2(n+1)(n+2)a_{n+2}x^{n}-\sum_{n=\alpha}^{\infty}(n+1)(n+2-\alpha)a_{n+2-\alpha}x^{n}=\sum_{n=0}^{\infty} a_nx^n$$
Finally equating these coefficients gives us the following recursion for $n\geq
\alpha$
$$\omega^2(n+1)(n+2)a_{n+2}-(n+1)(n+2-\alpha)a_{n+2-\alpha}=a_n$$
And for $n<\alpha$, we have
$$\omega^2(n+1)(n+2)a_{n+2}=a_n$$
So, all power series with sequences $a_n$ that satisfy these recursions will work. 
If anyone solves these recursions generally, I will be surprised to say the least.
A: Hint:
$(\omega^2-x^\alpha)\psi''-\alpha x^{\alpha-1}\psi'-\gamma\psi=0$
$(x^\alpha-\omega^2)\psi''+\alpha x^{\alpha-1}\psi'+\gamma\psi=0$
Which relates to an ODE of the form http://science.fire.ustc.edu.cn/download/download1/book%5Cmathematics%5CHandbook%20of%20Exact%20Solutions%20for%20Ordinary%20Differential%20EquationsSecond%20Edition%5Cc2972_fm.pdf#page=267.
