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2. INTRODUCTORY APPLICATIONS

In the previous chapter, we laid the foundations of singular perturbation theory and,
although we will need to add some specific techniques for solving certain types of
differential equations, we can already tackle simple examples. In addition, we will see
that we can apply these ideas directly to other, more routine problems—and this is
where we shall begin. Here, we will describe how to approach the problem of finding
roots of equations (which contain a small parameter), and how to evaluate integrals of
functions which are represented by asymptotic expansions with respect to a parameter.
Finally, we begin our study of differential equations by examining a few important,
fairly straightforward examples which are, nonetheless, not trivial.

2.1 ROOTS OF EQUATIONS
At some stage in many mathematical problems, it is not unusual to be faced with the
need to solve an equation for specific values of an unknown. Such a problem might
be as simple as solving a quadratic equation:

or finding the solution of more complicated equations such as