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to the solution. To proceed, let us be given, as an example, the specific data:

and we will find the first two terms in the solution valid away from the boundary layer,
and then the first terms in each boundary-layer solution. From (3.46a), we obtain

where A and B are arbitrary functions; the available data, (3.47), requires that

and so we have

The next term, satisfies (3.46b) with

Equation (3.46b) can be written

and so the solution for is

and we have the asymptotic solution

This two-term asymptotic expansion does not satisfy the given data on y = 0 or on
y = 1; thus we require (thin) layers near these two boundaries of the domain.
The first stage involves finding the size of the boundary layers; let us introduce
with as for the boundary layer near y = 0. Further, we note
that u = O(1) here and, of course, there is no scaling in x. Thus, with
equation (3.44) becomes