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64 2. Introductory applications

requires the analysis of the equation for with general, bounded and
which is possible, but beyond the aims of this text.)
As should be clear from this calculation, it is to be anticipated that special prop-
erties, relevant to a particular problem, may have to be invoked. Here, for example,
we took advantage of the underlying Riccati equation; other problems may require
quite different approaches. However, we must also emphasise that, for many practical
and important problems encountered in applied mathematics, these calculations are
often too difficult to succumb to such a general analysis. Indeed, the conventional
wisdom is that, if breakdowns have been identified, rescaling employed and asymp-
totic solutions found (and matched, as required), then we have produced a sufficiently
robust description. It should be noted that the process of rescaling might involve a
consideration of all possible scalings allowed by the governing equation, which will
then greatly strengthen our trust in the results obtained. Those readers who prefer the
more rigorous approach that such discussions afford are encouraged to study the texts
previously mentioned. In this text, however, we shall proceed without much further
consideration of these more formal aspects of the asymptotic solution of differential
equations.
Now that we have presented the salient features of the method of constructing
solutions, we apply it to another example.

E2.7 A regular second-order problem
We seek an asymptotic solution, as of

with and the primes here denote derivatives. First, we assume
that there is a solution, for some of the form

Thus we obtain

and so on, with

(if the expansion is valid at the end-points). The general solution of (2.23a) is