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and we will need to invoke the matching principle to decide which sign is appropriate.
Thus from the first term in (2.35), we see that

from (2.37) we obtain

and then (2.38) and (2.39) match only for the positive sign. (Note that, because y has
also been scaled, this must be included in the construction which enables the matching
to be completed.) The solution for the first term is therefore

and then the second term is obtained directly as

the resulting 2-term asymptotic expansion is

We have found that this problem, (2.34), requires an asymptotic expansion for
x = O(1) and another for In addition, it is clear that (2.40) does not
further break down as (and it is fairly easy to see that no later terms in
the expansion will alter this observation): two asymptotic expansions are sufficient.
The appearance of an algebraic problem implies that all solutions are the same—any
variation by virtue of different boundary values is lost for how is
this possible? The explanation becomes clear when (2.35) is examined more closely; the
terms associated with the arbitrary constants (at each order) are exponential functions,
and for these are all proportional to they are exponentially
small. Such terms have been omitted from the asymptotic expansion for if
they had been included, then the matching of these terms would have ensured that
information about the boundary values would have been transmitted to the solution
valid for albeit in exponentially small terms.