TABLE OF CONTENTS Few differential equations, whether linear or nonlinear, can be solved exactly in terms of a finite number of the elementary functions. Even for the case where exact solutions are obtained, they are either in implicit form or are so complex in mathematical structure that little useful information can be directly derived from them. Fortunately, for many equations modeling oscillatory behavior, a large body of calculational methods exist for determining analytical approximations to the required solutions.
The main purpose of this chapter is to examine some of these methods and apply them to differential equations modeling one-dimensional oscillatory systems. We begin with a brief discussion of the general philosophy of perturbation procedures, what such methods are required to do, and how they work in practice. This presentation is followed by sections giving the details for several important perturbation methods: first-order averaging, the Lindstedt-Poincar method, and harmonic balance. Following each of these individual sections, we illustrate the use of each method by applying it to a number of second-order, nonlinear differential equations.
Section 8.10 is devoted to the issues related to constructing an averaging procedure for a class of second-order, nonlinear difference equations corresponding to discrete models of nonlinear oscillators. The following section 8.12 applies this method to a number of equations.
A bibliography, at the end of the chapter, gives a selected listing of books on both the general theory of perturbation procedures and their application to the nonlinear differential equations arising in the analysis of particular systems in...
