ABSTRACT
To motivate the types of Sturm-Liouville problems, the authors present, without detailed derivations, a few important problems of mathematical physics and the Sturm-Liouville eigenvalue and boundary value problems to which they lead, usually via separation of variables in a partial differential equation. However, eigenvalue problems arise in many contexts involving ordinary and partial differential equations as well as in matrix theory and more general operator settings. The authors use physical arguments to motivate the existence of a Green's function, to point out some of its important properties, and to find the solution formula. A key step in separation of variables is to superpose the separated solutions with the aim of satisfying all the remaining conditions in the initial boundary value problem at hand. There are occasions when it is desirable or necessary to deal with complex-valued solutions to differential equations.
