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5. SOME WORKED EXAMPLES ARISING FROM PHYSICAL PROBLEMS

In this final chapter, the aim is to present a number of worked examples where most
of the details are given explicitly; what little is left undone may be completed by the
interested reader (although no formal exercises are offered). Also, we will not dwell
upon the purely technical aspects of finding the solution of a particular differential
equation. These examples are taken from, or based on, texts and papers that introduce,
describe, develop, explain and solve practical problems in various fields; references to
appropriate source material will be included. Most have arisen—not surprisingly—
from the physical sciences, but we have attempted to provide a fairly broad spread of
topics. Each problem is described with sufficient detail (we hope) to enable it to be
put into context, although it would be quite impossible to include all the background
ideas for those altogether unfamiliar with the particular field. To this end, the prob-
lems are collected under various headings (such as ‘mechanical & electrical systems,
‘semiconductors’ or ‘chemical & biological reactions’) and so the reader with particular
interests might turn to specific ones first. Nevertheless, the hope is that every problem
is accessible, as an example in singular perturbation theory, to those who have followed
this (or any other suitable) text. The technique adopted to construct the asymptotic
solution will be mentioned, and a reference will be given to a relevant section or
example from the earlier chapters of this text.
A number of the examples and exercises that have already been discussed have been
taken from various important applications; in some cases, those presented in this chap-
ter build on and expand these earlier problems. The reader should be aware, therefore,