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66 2. Introductory applications

2.4 ORDINARY DIFFERENTIAL EQUATIONS: SIMPLE SINGULAR PROBLEMS
Now that we have introduced the simplest ideas that enable solutions of differential
equations to be constructed, we must extend our horizons. The first point to record is
that, only quite rarely, do we encounter problems that can be represented by uniformly
valid expansions (although, somewhat after the event, we can often construct such
expansions—in the form of a composite expansion, for example; see §1.10). The
more common equations exhibit singular behaviour, in one form or another; the
simplest situation, we suggest, is when the techniques used above (§2.3) produce
asymptotic expansions that break down, resulting in the need to rescale, expand again
and (probably) invoke the matching principle. (Other types of singularity can arise, and
these will be described in due course.) To see how this approach is a natural extension
of what we have done thus far, we will present a problem based on the equation given
in (2.11).
We consider

for the important new ingredient here is the variable coefficient (which, we
note, is for x = O(1)). We seek a solution in the form

and we will need to find the terms and (at least) in order to include a
contribution from the new part of the coefficient. The equations for the are

and so on; the boundary condition requires that

In this problem, we should expect that evaluation of the expansion on x = 1 is
allowed—all terms are defined for x = O(1)—but we must anticipate difficulties as

The solutions for the functions and follow from the results given in
(2.15) and (2.16), respectively, but with the particular integral omitted; thus