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x = 1 (or, possibly, both)? Here, we will assume that there is a single boundary layer
near x = 0; the problem of finding the position of a boundary layer will be addressed
in the next section, at least for a particular class of ordinary differential equations. Let
us now return to equation (2.63).
As should be evident from example E1.4, and will become very clear in what follows,
it is the appearance of the small parameter multiplying the highest derivative that is
critical here. The presence of in another coefficient is altogether irrelevant to the
general development; it is retained only to allow direct comparison with E1.4. We seek
a solution of (2.63) in the form

and then we obtain

and so on. The only boundary condition available to us (because of the assumption
about the position of the boundary layer) is

Thus

and indeed, in this problem, we then have although exponentially
small terms would be required for a more complete description of the asymptotic
solution valid for x = O(1); so we have

The scaled version of (2.63) is obtained by writing for and
(because y = O(1) as although any scaling on Y will vanish identically
from the equation); thus

The relevant balance, as we have already seen in (2.62), is or
giving