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KdV equation, (3.21). Equation (3.78), like the KdV equation, is also an important
equation; it is the Burgers equation (Burgers, 1948) which can be solved exactly by
applying the Hopf-Cole transformation (Hopf, 1950; Cole, 1951) which transforms the
equation into the heat conduction (diffusion) equation. As with the corresponding
KdV problem (in E3.1), the matching condition is as and, for
suitable the asymptotic expansions (3.76) are uniformly valid as The
solution that we have obtained describes a weak pressure wave moving through the
near-field (t = O(1)) and, as t increases into the far-field, the wave-front steepens, but
this effect is eventually balanced by the diffusion (when Finally the wave
will settle to some steady-state profile—a profile which is regarded as a model for a
shock wave in which the discontinuity is smoothed; see Q3.5.

This concludes all that we shall present here, as examples of fairly routine singular per-
turbation problems in the context of partial differential equations. Further ideas—very
powerful ideas—which are applicable to both ordinary and partial differential equa-
tions will be developed in the next chapter. We complete this chapter on some further
applications by examining two rather more sophisticated problems that involve ordinary
differential equations. The first employs the asymptotic expansion in a parameter in
order to study an important equation in the theory of ordinary differential equations:
Mathieu’s equation. The second develops a technique, which is an extension of one of
our earlier problems associated with wave propagation, that enables the asymptotic so-
lution of certain ordinary differential equations to be written in a particularly compact
and useful form—indeed, one that exhibits uniform validity when none appears to
exist.

3.4 FURTHER APPLICATIONS TO ORDINARY DIFFERENTIAL EQUATIONS
The Mathieu equation, for x(t),

where and are constants, has a long and exalted history; it arose first in the work of
E-L. Mathieu (1835–1900) on the problem of vibrations of an elliptical membrane. It
also applies to the problem of the classical pendulum in which the pivot point is oscil-
lated along a vertical line, one result being that, for certain amplitudes and frequencies
of this oscillation (which corresponds to certain and the pendulum becomes
stable in the up position! It is also relevant to some problems in electromagnetic-wave
propagation (in a medium with a periodic structure), some electrical circuits with spe-
cial oscillatory properties and in celestial mechanics. The equation is conventionally
analysed using Floquet theory (see e.g. Ince, 1956) in which the solution
for in general, a complex constant, has y(t) periodic (with period or for
certain We will show how some of the properties of this equation are readily
accessible, at least in the case