In this chapter, and the following two, our main aim is to show how to use a knowledge of the eigenvalues and eigenvectors of in order to draw the phase diagram for the equation ẋ =x. As in Chapter 7, this phase diagram will illustrate the qualitative behaviour of the solutions by showing a representative choice of the curves traced out by the solutions (x(t), y(t)), labelled with an arrow to indicate in which direction the solution moves as t increases.

In each chapter we will examine one of the three possibilities (two distinct real eigenvalues, a complex conjugate pair of eigenvalues, or a repeated eigenvalue) and for each case we will show

1. (i) how an appropriate change of coordinates, based on the eigenvectors of the matrix, can be used to transform the differential equation into a standard, simpler (canonical) form;

2. (ii) how to find the explicit solution of this simple form of the equation;

3. (iii) how to draw the phase portrait for the simple equation;

4. and hence

5. (iv) how to find the explicit solution of the original equation; and

6. (v) how to draw its phase portrait.

Since we already have a reliable method for solving coupled linear equations, (ii) and (iv) will be much less important than (iii) and (v).

The aim of this book is to deal with all of the elementary methods for obtaining explicit solutions of ordinary differential equations, and then to introduce the ideas of qualitative analysis using phase plane techniques. Simple difference equations are also included, since their methods of solution are similar to those for linear differential equations. As well as being, I hope, an internally consistent choice of material, this selection of topics also has the advantage of preparing a student for a basic course on dynamical systems.

The book arose from my unsuccessful efforts to find a suitable text to recommend when I taught the first year Warwick differential equations course. Although there are a number of well-established and successful textbooks that treat this subject (these are discussed, along with other possibilities for further reading, in the final chapter), they seem either to include a large amount of additional material, or to concentrate only on the more advanced topics. I therefore produced a detailed set of lecture notes, which, with the encouragement of Alan Harvey and David Tranah, and most significantly Kenneth Blake at Cambridge University Press, eventually became this book. My thanks here to all those students who made useful suggestions while this book was still at the lecture note stage.

Part I contains an informal discussion of the issues of existence and uniqueness of solutions, and treats the standard classes of first order differential equations that can be solved explicitly, as well as covering exact equations and substitution methods.

Differential equations date back to the mid-seventeenth century, when calculus was discovered independently by Newton (c. 1665) and Leibniz (c. 1684). Modern mathematical physics essentially started with Newton's Principia (published in 1687) in which he not only developed the calculus but also presented his three fundamental laws of motion that have made the mathematical modelling of physical phenomena possible.

Historically, advances in the theory of differential equations have come from the insights gained when trying to treat specific physical models. Despite this somewhat piecemeal development, the subject has become a well-defined and coherent area of mathematics. This book adopts a theoretical point of view, developing the theory to the point at which it can no longer be described as ‘basic differential equations’ and is about to become entangled with more advanced topics from the theory of dynamical systems. Of course, applications are used throughout to serve as motivation and illustration, but the emphasis is on a clean presentation of the mathematics.

You may find that some of the problems covered in the first few chapters are already familiar. The methods of solving these problems are well established, and you may be well practised at applying them. However, we will take care here to show why these methods work; giving proper justification of the methods can take some time, but as mathematicians we should not be satisfied merely with a set of ‘recipes’.