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78 2. Introductory applications

(Note that we have used the identity

and similarly for the second derivative; this is valid for For general A and B,
(2.59) implies that y = O(1) as so we select

and no balance exists (that is, between and as Again we deduce, on
the basis of this scaling argument, that the asymptotic expansion is uniformly valid for
(exactly as we found was the case for expansion (2.25)). The special case
in which produces as and so we now require
but any balance is still impossible.

E2.12 Scaling procedure applied to a new equation
For our final example, we consider the equation

where r (x) is either zero or r (x) = x; the two boundary conditions are either one at
each end of the domain, or both at one end—it is immaterial in this discussion. The
general solution of the dominant terms in (2.60) as for x = O(1), is

the latter applying when r (x) = x. The general scaling in the neighbourhood of any
is and which gives

for the equation with Because this equation is linear and homogeneous, the
scaling in y is redundant: it cancels identically. (We may still require to measure
the size of y, but it can play no rôle in the determination, from the equation, of any
appropriate scaling near The balance of terms, as requires either that
or that and only the latter is consistent, so the
former balances the terms and Y, but then is the dominant contributor! Thus
any scaled region must be described by although this analysis cannot
help us decide if an exists, or what might be; we will examine this issue in the
next section.