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88 2. Introductory applications

(2.85) when expressed in terms of x (as will be necessary for any matching); this gives

But we are seeking solutions that remain bounded as and this is possible only
if the exponent in (2.87) is non-positive for Thus we require

and this means that for the same xs i.e. (for otherwise x < C can occur
in the domain, and the integral will change sign). Then, for x sufficiently close to
we have

and we have a choice for the boundary-layer variable. Hence, for a (x) > 0, the bound-
ary layer must sit near the left-hand edge of the domain. Conversely, the same argument
in the case a (x) < 0 requires that the boundary layer be situated near (the right-
hand edge of the domain). When we apply this rule to equation (2.63):
we see that and so the boundary layer is
in the neighbourhood of x = 0 (and we introduced for this example).
Similarly, equation (2.77): has a(x) = – 1 < 0,
and so the boundary layer is now near x = 1 (and we used
It is rarely necessary to incorporate the formal definition of g(x) to generate the
appropriate variable that is to be used to represent the boundary layer (although it
will always produce the simplest form of the solution). For example, the equation of
this class: has a boundary layer near x = 2
(because a (x) = –1/(2 + x) < 0 for Now an appropriate scaled variable is
simply giving

and this choice will suffice, even though higher-order terms will require the expansion
of (but this is usually a small price to pay—and we already know that
this asymptotic expansion will breakdown for so retaining in the
coefficient has no unforeseen complications). It should also be noted that, exceptionally,
a boundary-layer-type problem may not require a boundary layer at all, in order to
accommodate the given boundary value (to leading order or, possibly, to all orders).
This is evident for the equation given in (2.63) (and see also the exact solution, (1.22));
in this example, if the boundary values satisfy the special condition then no
boundary layer whatsoever is required. Note, however, that if then