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and then (2.23b) becomes

which, in turn, has the general solution

where are the new arbitrary constants. The functions and are clearly
defined for and there is no suggestion of a breakdown, so we impose the
boundary conditions (2.24) to give

and then our asymptotic expansion (to this order) is

Now that we have obtained the expansion, (2.25), we are able to confirm that we
have a 2-term uniformly valid representation of the solution. In order to examine the
general term in this asymptotic expansion, if this is deemed necessary, we can follow
the method described earlier. Thus we may write

where, in particular, we have and for the general
solution for is

where and are determined to satisfy the two boundary conditions. The essentials
of the argument are then as we have already outlined in our first, simple, presentation:
is bounded (on [0, 1]), so is and hence so is etc., for all Further,
as and as for the asymptotic expansion is uniformly
valid.

Some further examples of regular expansions can be found in Q2.9 and 2.10, and an
interesting variant of E2.7 is discussed in Q2.15.