I'm trying to understand how to solve PDEs that are Sturm Liouville problems. I've read a couple of presentations about this, but I'm lost as to the implications for the solution process. Most descriptions discuss putting the second order differential equation into a Sturm-Liouville form. From this there are a set of implications that I can prove about ordered eigenvectors, the sign of the eigenvectors, the orthogonality of eigenfunctions, etc. While I understand these implications, I don't understand how this helps me solve a Sturm-Liouville problem. Additionally, the examples I have seen are the most basic (like the trivial case where p=1 q=0 and r=1), and they all begin with solving the ODE. I don't understand if this is how you solve EVERY Sturm-Liouville problem, or if this is just done for the sake of demonstration of a connection? I'm concerned with how to do cases like p=x, q=0, r=1/x and boundary conditions u(1)=u'(e)=0. Should I be considering the ODE solution when solving this, or do something else?