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if the domain is then from (2.56) we see that y = O(1) as (unless
; see below). So we select (2.57) becomes

and there is no choice of scaling, as which balances the term against
We conclude that a second asymptotic region does not exist, and hence that the
expansion for x = O(1) is uniformly valid on this (bounded) domain, which agrees
with the discussion following (2.17). (In the special case as
and so a matched solution now requires producing

again, there is no choice of as which balances against
On the other hand, if the domain is then as and we
require for a matched solution to exist; equation (2.57) now becomes

This time, with (because the O(1) terms balance if
e.g. or which recovers the scaling used to give (2.40) in E2.8.

E2.11 Scaling for problem (2.22)
Consider the equation

with and suitable boundary conditions (which may both be at one end,
or one at each end). The general solution of the dominant terms from (2.58), with
x = O(1) as is

(as used to generate (2.25)). In this example, the asymptotic expansion, of which (2.59)
is the first term, may break down as or as or, just possibly, as
with All these may be subsumed into one calculation by introducing a
simple extension of our method of scaling: let where we may allow
in this formulation. Then with the usual (2.58) becomes