Abstract
An elementary course in ordinary differential equations often leaves the impression that only initial value problems are of interest. There are a variety of other possible conditions that are important in applications. For initial value problems, all the conditions are placed at one value of the independent variable—the initial time. When conditions occur for different values of the independent variable, the resulting problem is called a boundary value problem. This chapter discusses boundary value problems. It presents second-order linear boundary value problems and describes the development of a general structure of such problems. It also presents a linear second-order differential equation with variable coefficients and with conditions placed on the solution at two different values of the independent variable. Although the theory of linear boundary value problems can be developed in this generality, computations are really possible at this level only when the coefficients are constant. The chapter also presents the idea of a Green's function and discusses the treatment of inhomogeneous problems. The key question for the existence of such functions turns out to be the distance between zeros of certain solutions.
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