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where is an arbitrary constant. Then, from (2.13b), we have

which can be written (on using the integrating factor as

This produces the general solution

where is a second arbitrary constant. It is immediately clear that these first two terms
in the expansion are defined (and well-behaved i.e. no hint of a non-uniformity) for
so we may impose the boundary conditions, (2.14a,b); these produce

Thus our asymptotic expansion, so far, is

and this is certainly uniformly valid for we have a 2-term expansion of the
solution. (Note that the specification of the domain is critical here; if, for example,
we were seeking the solution with the same boundary condition, but in then
(2.17) would not be uniformly valid: there is a breakdown where i.e.
see the problem in (2.34), below.) The evidence in (2.17) suggests that
we have the beginning of a uniformly valid asymptotic expansion i.e. (2.12) is valid
for and for (and it is left as an exercise to find and to check that
the inclusion of this term does not alter this proposition).
In order to investigate the uniform validity, or otherwise, of (2.12), one approach is
to examine the general term in the expansion; this is the solution of

where with The solution to (2.18)
is

but and are bounded functions for and hence so is and
then so is and hence all the In particular, as and
constants) as there is no breakdown of the asymptotic