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94 2. Introductory applications

which has the general solution

and requires (It should be noted that as
which, alone, indicates a breakdown where this, as we shall see
below, is irrelevant).
For the boundary layer, we scale with then the equation,
(2.101), becomes

and so we must select which leads to

(The apparent scaling, is therefore redundant—it is smaller than that required
in the boundary layer.) We seek a solution with

and the available boundary condition, then gives

Finally, we determine by invoking the matching principle; from (2.102) we obtain
and from (2.103) we see that

which match if the boundary-layer solution is therefore

where k is given in (2.104).