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We consider the equation

for with and (where and are independent of
for suitable x = O(1), we seek a solution in the usual form

which gives

and so on. The general solution to equation (2.108a) is

we exclude the solution because we will consider problems for which
and (Any special solutions which may need to make use of the zero solution
are easily incorporated if required.) The next term in this asymptotic expansion is
obtained from (2.108b) i.e.

which yields

where is the second arbitrary constant (and we have taken
It is clear, however, that it is impossible to proceed without more information about
the boundary values, and Let us examine, first, the problem for which both
values are positive; we therefore assume that a solution, y > 0, exists and hence that
any boundary layer must be situated in the neighbourhood of x = 0 (indicated by the
term with y > 0). With this in mind, we may use the one available boundary
condition away from x = 0, i.e. thus we obtain

Correspondingly, with we see that we will assume hereafter that
(but we clearly have an interesting case if for then the
solution does have a zero near even with possibility not pursued
here).