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The function is periodic in T only if

thus, with (4.23), we obtain

(Note that, if we allow then and which corresponds to
the results obtained in Q3.24; the change of sign in is because here the leading
term involves cos rather than sin.) This leaves the solution of (4.24) as

with

The analysis of equation (4.20c), for follows the same pattern (and see
also E4.1), but here the details are considerably more involved; we will not pur-
sue this calculation any further (for we learn nothing of significance, other than to
show that which is left as an exercise). The solution, to this order, is
therefore

Other examples based on small adjustments to the equation for a linear oscillator can
be found in exercises Q4.1–4.9. It might be anticipated, in the context of oscilla-
tors governed by ordinary differential equations, that the method of multiple scales
is successful only if the underlying problem is a linear oscillation i.e. controlled by an
equation such as this would be false. In an important extension of these
techniques, Kuzmak (1959) showed that they work equally well when the oscillator
is predominantly nonlinear. Of course, the fundamental oscillation will no longer be
represented by functions like sin or cos, but by functions that are solutions of nonlinear
equations e.g. the Jacobian elliptic functions.

4.2 NONLINEAR OSCILLATORS
Equations such as

possess solutions that can be expressed in terms of sn, cn or dn, for example. (In the
Appendix we present all the basic information about these functions that is necessary for