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The leading-order problem, as is therefore (with equations
(5.39), (5.40) and (5.41), and (5.42a,b) replaced by

The solution for can be found (by using the Laplace transform again) and
then used to determine some details of this calculation are given in Crank (1984).
Although the form of is cumbersome, the resulting expression for on
is very straightforward, yielding

which matches precisely with (5.31) when this is expanded for The difficulties
in as have been overcome.

This example has required us to undertake some quite intricate analysis in terms of
singular perturbation theory, coupled with a careful appreciation of the details of the
physical processes involved. This problem, perhaps more than the previous three, shows
how powerful these techniques can be in illuminating the details. We now turn to a
far more routine type of calculation, although the equation and physical background
are important, and the resulting solution has far-reaching consequences.

E5.5 The van der Pol/Rayleigh oscillator
This classical example requires a fairly routine application of the method of multiple
scales to a nearly linear oscillator (cf. E 4.2), although the solution that we obtain takes a
quite dramatic form. The equation first came to prominence following the work of van
der Pol (1922) on the self-sustaining oscillations of a triode circuit (for which the anode
current-voltage law takes the form of a cubic relation). However, essentially the same
equation had already been discussed by Rayleigh (1883), as a model for ‘maintained’
vibrations in, for example, organ pipes. (A simple transformation takes Rayleigh’s equa-
tion into the van der Pol equation.) We will write the equation in the Rayleigh form

for in the context of the van der Pol problem, is proportional to the grid
voltage, V. [A circuit diagram is given in figure 11, and the governing equations for
this triode circuit are

where and are positive constants.]