Overview

In an ordinary differential equation (ODE), there is a single independent variable. Commonly ODEs model change over time, so the independent variable is t (time). Our interest in ODEs derives from the following fact: both the Fourier series method and the finite element method reduce time-dependent PDEs into systems of ODEs. In the case of the Fourier series method, the system is completely decoupled, so the "system" is really just a sequence of scalar ODEs. In Section 4.2, we learn how to solve the scalar ODEs that arise in the Fourier series method.

The finite element method, on the other hand, results in coupled systems of ODEs. In Section 4.3, we discuss the solution of linear, coupled systems of first-order ODEs. Although we present an explicit solution technique in that section, the emphasis is really on the properties of the solutions, as the systems that arise in practice are destined to be solved by numerical rather than analytical means. In Sections 4.4 and 4.5, we introduce some simple numerical methods that are adequate for our purposes.

We close this chapter by interpreting our simple solutions in terms of Green's functions. Although we do not emphasize the method of Green's function in this book, we do explain the basic idea in Section 4.6.

4.1 Converting a higher-order equation to a first-order system

We begin our discussion of ODEs with a simple observation: It is always possible to convert a single ODE of order two or more to a system...