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(You should confirm that, in the above, both and cancel identically, to this order.)

Further examples that make use of these ideas can be found in exercises Q2.6, 2.7 and
2.8. With a little experience, it should not be too difficult to recognise how many
terms need to be retained in each expansion in order to produce to a desired
accuracy. The region that gives the dominant contribution is usually self-evident,
and quite often this alone will provide an acceptable approximation to the value of
the integral. Furthermore, terms that contain the overlap variables can be ignored
altogether, because they must cancel (although there is a case for their retention—
which was our approach above—as a check on the correctness of the details).

2.3 ORDINARY DIFFERENTIAL EQUATIONS: REGULAR PROBLEMS
We now turn to an initial discussion of how the techniques of singular perturbation
theory can be applied to the problem of finding solutions of differential equations—
unquestionably the most significant and far-reaching application that we encounter.
The relevant ideas will be developed, first, for problems that turn out to be regular
(but we will indicate how singular versions of these problems might arise, and we will
discuss some simple examples of these later in this chapter). Clearly, we need to lay
down the basic procedure that must be followed when we seek solutions of differential
equations. However, these techniques are many and varied, and so we cannot hope
to present, at this stage, an all-encompassing recipe. Nevertheless, the fundamental
principles can be developed quite readily; to aid us in this, we consider the differential
equation

for This problem, we observe, is not trivial; it is an equation which, although
first order, is nonlinear and with a forcing term on the right-hand side.
The first stage is to decide on a suitable asymptotic sequence for the representation
of Here, we note that the process of iteration on the equation, which can be