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62 2. Introductory applications

expansion. The problem posed in (2.11) is therefore regular, resulting in a uniformly
valid asymptotic expansion.
More complete, formal and rigorous discussions of uniform validity, in the context
of differential equations, can be found in other texts, such as Smith (1985), O’Malley
(1991) and Eckhaus (1979). Typically, these arguments involve writing

where and then showing that remains bounded for
and for We will outline how this can be applied to our problem,
(2.11); first, we obtain

with for Since each satisfies an appropriate differential
equation and boundary condition, this gives

where comprises the terms, and smaller, from the expansion of (after
division by A uniform asymptotic expansion requires that is bounded as
for and To prove such a result is rarely an elementary exercise
in general, and it is not trivial here, although a number of approaches are possible.
One method is based on Picard’s iterative scheme (which is a standard technique
for proving the existence of solutions of first order ordinary differential equations in
some appropriate region of (x, y)-space); this will be described in any good basic text
on ordinary differential equations (e.g. Boyce & DiPrima, 2001). Another possibility,
closely related to Picards method, is formally to integrate the equation for
thereby obtaining an integral equation, and then to derive estimates for the integral
term (and hence for We will outline a third technique, which involves the
construction of estimates directly for the differential equation, and then integrating a
reduced version of the equation for
At this stage we do not know if is of one sign, for or if it changes
sign on this interval; however, we may proceed without specifying or assuming the
nature of this property, but it will affect the details; first we write

But we do know that each is bounded (for and hence so is which
we will express in the form (a constant independent of and so