TABLE OF CONTENTS In this book we have covered all of the basic methods for finding the explicit solutions of simple first and second order differential equations, along with some qualitative methods for coupled nonlinear equations. We have also discussed difference equations, and seen how complicated the dynamics of even very simple iterated nonlinear maps can become.
There are two ways in which to proceed further with the material developed here. One arises from turning first to the study of partial differential equations, while the other essentially continues from where we have left off.
Partial differential equations model systems that have spatial as well as temporal structure, for example the temperature throughout an object, the vibrations of a string or a drum, or the velocity of a fluid.
In general linear partial differential equations are easier to solve. By using the technique known as separation of variables it is possible to convert such a problem into an ordinary differential equation. This was touched on briefly in Exercise 20.10, and the exercises in this chapter apply this method in more detail for the example of the vibrating string.
Viewed in general this approach leads to the theory of Fourier series (the expansion of an arbitrary function as a sum of sines and cosines, see Exercises 38.7 and 38.8), and its extension via the Sturm Liouville theory of boundary eigenvalue problems
These topics are treated at length in many differential equation textbooks, for example,
W. E. Boyce and...
