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82 2. Introductory applications

We seek a solution of this equation in the usual form:

which gives

and so on. We have available the one boundary condition prescribed at x = 0 (which
is in the region of the boundary layer) i.e. at X = 0, so

From (2.67a) and (2.68) we obtain

where is an arbitrary constant, and then (2.67b) becomes

Thus

where and are the arbitrary constants, which must satisfy from
this gives the solution

The 2-term asymptotic expansion valid for is therefore

which is to be matched to (2.65); this should uniquely determine and We write
(2.65) as

retaining terms O(1) and (as used in (2.70)); correspondingly, we write (2.70) as