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102 2. Introductory applications

and then (exactly as described for we obtain

We now determine the constants and by matching (2.123) with, in
turn, (2.124) and (2.125). First, from (2.123) with and we obtain

from (2.124) we have

which requires and then Again, from (2.123), but now with
and we obtain

and, finally, (2.125) gives

Now we require and thus, collecting all these results, we see
that

and hence, to leading order, we have

with

So, indeed, boundary layers are required at each end in order to accommodate the
boundary conditions there (although we may note that the solution for does
satisfy but not the derivative conditions).

Some further examples of higher-order equations that exhibit boundary-layer be-
haviour are offered in Q2.26.
This chapter has been devoted to a presentation of some of the fairly routine ap-
plications of singular perturbation theory to various types of mathematical problem.