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98 2. Introductory applications

Now, as we see that and if this does not equal
then a boundary layer is required near x = 0. For this layer, we write with
(= O(1)), and then (2.107) becomes

with the choice We seek a solution where

and we have chosen to write the arbitrary constant of integration as any
other choice produces a solution which cannot be matched—an investigation that is
left as an exercise. The next integral of (2.113) gives the general solution

and to satisfy this can be written as

The value of the remaining arbitrary constant, is now determined by matching
(2.114) and (2.111); from (2.111) we obtain

and from (2.114) we see that

which requires that Thus we have successfully completed the initial cal-
culations in the construction of asymptotic expansions valid for x = O(1) and for
these demonstrate that, in the case and we have a solution
which satisfies y > 0 for

E2.19 A problem which exhibits either a boundary layer or a transition
layer: II
We repeat example E2.18, but now with the boundary values (chosen to avoid
the difficulties already noted) and and thus the solution must change sign (at
least once) in order to accommodate these boundary values. This indicates the need