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76 2. Introductory applications

these are

where ‘ – ’ denotes terms used, in the first approximation, with x = O(1), and de-
notes, correspondingly, the terms used where (The derivative has not
been labelled, but it will automatically be retained, by virtue of the multiplication
of terms, when labelled terms are used in an approximation.) The important inter-
pretation is that some terms—here all—used where x = O(1) are balanced against
some terms not used previously (in the first approximation), but now required where
When we impose this requirement on (2.54), and note that the breakdown
is as i.e. then the only balance occurs when we choose
or, because we may define in any appropriate way, simply It is impossi-
ble to balance the term in against the O(1) terms here (to give different leading
terms) because, when we do this, the dominant term then becomes
which is plainly inconsistent. Note that and as can never be
balanced.
Thus, armed only with the general scaling property, the behaviour y ~ 1/x as
and the requirement to balance terms, we are led to the choice this
does not involve any discussion of the nature of the breakdown of the asymptotic
expansion. This new procedure is very easily applied, is very powerful and is the most
immediate and natural method for finding the relevant scaled regions for the solution
of a differential equation. We use this technique to explore two examples that we have
previously discussed, and then we apply it to a new problem.

E2.10 Scaling for problem (2.11) (see also (2.34))
Consider the equation

with a boundary condition given at where as the
domain is either or The solution of the first term in the asymptotic
expansion valid for x = O(1) is (see (2.15))

The general scaling, X, in (2.55) gives