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1. MATHEMATICAL PRELIMINARIES

Before we embark on the study of singular perturbation theory, particularly as it is rele-
vant to the solution of differential equations, a number of introductory and background
ideas need to be developed. We shall take the opportunity, first, to describe (without
being too careful about the formalities) a few simple problems that, it is hoped, explain
the need for the approach that we present in this text. We discuss some elementary dif-
ferential equations (which have simple exact solutions) and use these—both equations
and solutions–to motivate and help to introduce some of the techniques that we shall
present. Although we will work, at this stage, with equations which possess known
solutions, it is easy to make small changes to them which immediately present us with
equations which we cannot solve exactly. Nevertheless, the approximate methods that
we will develop are generally still applicable; thus we will be able to tackle far more
difficult problems which are often important, interesting and physically relevant.
Many equations, and typically (but not exclusively) we mean differential equations,
that are encountered in, for example, science or engineering or biology or economics,
are too difficult to solve by standard methods. Indeed, for many of them, it appears
that there is no realistic chance that, even with exceptional effort, skill and luck, they
could ever be solved. However, it is quite common for such equations to contain
parameters which are small; the techniques and ideas that we shall present here aim to
take advantage of this special property.
The second, and more important plan in this first chapter, is to introduce the ideas,
definitions and notation that provide the appropriate language for our approach. Thus