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When learning about Sturm-Liouville operators and their properties, we also learned that any second order linear differential operator can be written in Sturm Liouville form after multiplying by an appropriate weight function. We solved some problems using this approach, and the key reasons for using sturm liouville operators seemed to be the orthogonality of the eigenfunctions, and that they form a complete basis. However these are only properties of Sturm Liouville operators because they are self adjoint (glossing over the boundary requirements needed for this to be the case). So this begs the question as to whether all second order linear differential operators that are self-adjoint are necessarily in Sturm-Liouville form? And, if not, why bother with putting the operators in this form anyway?

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    $\begingroup$ Yes, the Sturm-Liouville form is a canonical form for a symmetric second order linear ODE, when the symmetry involves a weight function in the integral. By the way, selfadjoint arose out of Sturm-Liouville. Not the other way around. The study of symmetric matrices came out of the study symmetric ODEs of Sturm-Liouville type! Inner product arose out of Sturm-Liouville, as did eigenvalue/eigenvector analysis. $\endgroup$ – DisintegratingByParts Apr 3 '18 at 16:37

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