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2 1. Mathematical preliminaries

we will describe : order, asymptotic sequences, asymptotic expansions, expansions with
parameters, non-uniformities and breakdown, matching.

1.1 SOME INTRODUCTORY EXAMPLES
We will present four simple ordinary differential equations–three second-order and
one first-order. In each case we are able to write down the exact solution, and we will
use these to help us to interpret the difficulties that we encounter. Each equation will
contain a small parameter, which we will always take to be positive; the intention
is to obtain, directly from the equation, an approximate solution which is valid for
small

E1.1 An oscillation problem
We consider the constant coefficient equation

with x(0) = 0, (where the dot denotes the derivative with respect to t); this
is an initial-value problem. Let us assume that there is a solution which can be written
as a power series in

where each of the is not a function of The equation (1.1) then gives

where we again use, for convenience, the dot to denote derivatives. We write (1.3) in
the form

and, since the right-hand side is precisely zero, all the must vanish; thus
we require

(Remember that each does not depend on
The two initial conditions give