Consider the following eigenvalue problem
\begin{equation} \beta xf'(x)+\alpha f'(x)+\alpha xf(x)=Ef(x) \end{equation}
I would like to solve for $f(x)$ and $E$.
I know one set of solutions. $f(x)=e^{-\frac{\alpha}{\beta}x}$ and $E=-\frac{\alpha^{2}}{\beta}$. I was wondering if other solutions exist.