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166 4. The method of multiple scales

the calculations that we describe.) We discuss an example, taken from Kuzmak (1959),
which exemplifies this technique.

E4.3 A nonlinear oscillator with slowly varying coefficients
We consider the problem

for together with some suitable initial data; this is a Duffing equation with
slowly varying coefficients. The equation clearly implies that the slow scale should be
but what do we use for the fast scale? Here, we define a general form of fast
scale, T, by

where is to be determined, and the period of the oscillation in T is defined
to be a constant—an essential requirement in the application of multiple scales in
this problem. (In some more involved problems, it might be necessary to introduce
Equation (4.27) is transformed according to

which gives, with

and we seek a solution

where each is periodic in T. (The boundedness and uniform validity, that we aim
for as will depend on the particular we will assume that these
functions allow this.) Now (4.30) in (4.29) yields the sequence of equations

as far as Note that, although the first equation—for —is necessarily
nonlinear, all equations thereafter are linear.