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.(adsbygoogle = window.adsbygoogle || []).push({});

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?

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# Sturm-Liouville Core Concepts

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