I am trying to solve a regular Sturm-Liouville type Ordinary Differential Equation (ODE) with the following form:
\begin{equation} \frac{d}{dy} [q(y) \frac{dW(y)}{dy}]+p(y) W(y) = -\lambda W(y) \end{equation}
It satisfies all regular condition, $i.e.$, $q(y)>0$ between defining domain $(y_1, y_2)$. Meanwhile $p(y)$ is a very complicated, possibly with known parameter, rational function. $q(y)$ is a polynomial with order 3. While trying to do using ordinary power expansion (which is already very complicated, even using software), I am also thinking if I could use some special intergral transformation to simplify the equation. So, what would be the most practical transformation to use for such equation?
Also I would very much like to hear if there are any other ways to solve such kind of eigenvalue problem.