Page 185 -

168 4. The method of multiple scales

initial data will determine both this constant and Then, for known and suitable
and equations (4.34a,b) and (4.38) enable the complete description of the
first term in the asymptotic expansion (4.30).

This example has demonstrated that we are not restricted to nearly-linear oscillations,
although we must accept that mathematical intricacies, and the required mathematical
skills, are rather more extensive here than in the two previous examples. In addition,
problems of this type, because they are strongly nonlinear, often force us to address other
difficulties: we have used a periodicity condition, (4.37), but this fails if the solution
is not periodic—and this can happen. If the solution evolves so that then the
periodicity is lost because the period becomes infinite in this limit. In this situation, it
is necessary to match the solution for m = 1 to the periodic solution approximated as
an example of this procedure can be found in Johnson (1970). Some additional
material related to this topic is available in Q4.10 & 4.11.

4.3 APPLICATIONS TO CLASSICAL ORDINARY DIFFERENTIAL EQUATIONS
The method of multiple scales is particularly useful in the analysis of certain types of
ordinary differential equation which incorporate a suitable small parameter. We will dis-
cuss three such problems, the first of which we have already encountered: the Mathieu
equation(§3.4 and E3.6). The next two involve a discussion of a particular class of
problems—associated with the presence or absence of turning points (see §2.8)—with
a solution-technique usually referred to as WKB (or, sometimes, WKBJ); we will write
more of this later.
The Mathieu equation, discussed in §3.4 is

to which we will apply the method of multiple scales. However, before we undertake
this, we need to know what the fast and slow scales should be; this requires a little care.
Let us consider the equation with (fixed independent of

then we could select The equation for becomes

In general, we find that each has a particular integral proportional to
unless and then we have particular integrals that grow
in t. This condition will occur for and so the critical values of are
(n = 0, 1, 2 ...). From these points on the will emanate the transitional curves,