The Sturm-Liouville Problem

This chapter aims to develop a powerful approach, the method of separation of variables, which is efficient for solving all three types of Partial Differential Equations equations (ODEs). That method includes solution of a certain kind of problems for ordinary differential equations, the Sturm-Liouville problem, which is not always included in a standard course of ODEs. The Sturm-Liouville problem consists of a linear, second order ordinary differential equation containing a parameter whose value is determined by the condition that there exists a nonzero solution to the equation which satisfies a given boundary conditions. The set of orthogonal functions generated by the solutions to such problems gives the base functions for the Fourier expansion method of solving partial differential equations. The chapter discusses the properties of eigenvalues and eigenfunctions of a Sturm-Liouville problem.