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264 5. Some worked examples arising from physical problems

in the solution for y, but not for x. A suitable boundary-layer variable is
(rather than simply and we will take the opportunity to use the method of
multiple scales here, using T = t and (see §4.6). We introduce
and and then equations (5.152) become

with

The asymptotic solutions are expressed in the usual way:

which gives immediately that only; then we obtain

and so on.
In (5.155a) we choose to make the as simple as possible—the real
purpose behind the introduction of f(T) = f(t)—and so we write

and then we have simply

This is solved with ease, to produce

where is an arbitrary function. The initial conditions, (5.154), require that