I want to solve the following fourth-order differential equation in $G(x)$: $$ G^{(IV)}(x)=p(x)G(x) $$ If $p(x)=C\in\mathbb{R}$, the answer is a linear combinations of $\sin(x)$, $\cos(x)$ and $e^{\pm x}$. For general $p(x)$ I do not know.
I would be in particular interested in the cases of $p(x)$ being $p(x)=x^\chi$ or $p(x)=x^\chi e^{-x^2}$, with $\chi\ge 0$.
I thought about a WKB or Modified Airy Function approach, but I do not see a clear path.