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70 2. Introductory applications

At O(1) we have the basic oscillator; at the variable frequency; at the non-
linearity; at the damping. Thus, in order to investigate the leading contributions
(at least) to each of these properties of the oscillation, the asymptotic expansion must
be taken as far as the inclusion of terms (There is no suggestion that each will
necessarily lead to a breakdown, and an associated scaling, but each needs to be ex-
amined.) One further important observation will be discussed in the next section; we
conclude this section with two examples that exploit all these ideas.
E2.8 Problem (2.11) extended
We consider the problem

which is the same equation and boundary condition as we introduced in (2.11), but
now the domain is The asymptotic solution for x = O(1), which satisfies the
boundary condition, is (2.17) i.e.

and this breaks down where or Thus we introduce
but for this size of x, we observe that and so this also must be scaled: we
write

Equation (2.34) becomes

and we seek a solution

which gives

and so on. This result may cause some surprise: this sequence of problems is purely
algebraic—there is no integration of differential equations required at any stage.
Equation (2.36a) has the solution