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The solution of (3.98), with (3.94), produces exactly the solution obtained earlier,
(3.88), but now expressed in terms of rather than x:

We now turn to the vexing issue of solving (3.93)—and it is vexing because this is one
equation in two unknown functions: and How do we proceed?
The aim of this new technique is to obtain a uniformly valid asymptotic expansion—
if that is at all possible—for 1], although we have yet to determine
(which corresponds to x = 0). If there is to be any chance of success, then we must
remove any terms that generate non-uniformities in the asymptotic expansion for
from Q3.18, it is clear that the only such term here is i.e.
Thus we define so as to remove this term from the equation for it is
sufficient to remove such singular terms in any suitable manner, but if it is possible
to choose so as to leave an homogeneous equation for then this is the
usual move. (Other choices produce different asymptotic representations of the same
solution, but all equivalent to a given order in Here, therefore, we elect to write
the equation for as

leaving

An immediate response to this procedure is to observe that the term that causes all
the difficulty, has now appeared as a forcing term in the equation for —so
all we have succeeded in doing is moving the non-uniformity from one asymptotic
expansion to another! As we shall see, this is indeed the case, but the non-uniformity
in the expansion for the strained coordinate is not as severe as that in the expansion for
y. Before we address this critically important issue, we may note, from (3.97), that

where B is an arbitrary constant; but from (3.94) we see that we require B = 0 and so
Further, if this same strategy for selecting the equations for each is adopted,
then for every n = 1, 2, ..., and the exact solution, in terms of becomes

It is typical of these problems that it is not necessary to solve completely the equations
for each it is sufficient to examine the nature of the solutions as Thus we
will employ the same approach as described in Q3.18. First, from (3.92), we substitute