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142 3. Further applications

where C and D are arbitrary constants. Finally, from (3.83c), we have the equation

for which periodic solutions, require (and D = 0). Thus, to this
order, the transitional curve for

it is left as an exercise to show that, with the choice then

In summary, we have the stability boundaries given by

Set as an exercise, the case n = 2 can be found in exercise Q3.15. A corresponding cal-
culation to those described here, but now formulated in a way consistent with Floquet
theory, is set in Q3.16. More information about Mathieu equations and functions, and
their applications, is available in the excellent text: McLachlan (1964).
For our final discussion in this chapter, we will incorporate the idea introduced at
the end of E3.2, namely, a ‘correct’ solution in the ‘wrong place’, but applied here
to ordinary differential equations. We will describe the method of strained coordinates,
which has a long history; it was used first, in an explicit way, by Poincaré (1892), but
other authors had certainly been aware if the idea earlier, in one form or another.
Some authors refer to this as the PLK method after Poincaré, Lighthill (1949) and Kuo
(1953). The idea is exactly as mentioned above: the solution
(say), which is not uniformly valid, is made so by writing

where is a suitably chosen strained coordinate. Of course, only relatively special
problems have solutions that possess this structure, but it is regarded as a significant
improvement—over matched expansions—when it occurs. Indeed, we have met a
problem of this type in Chapter 1: E1.1, our very first example. There we found that
a straightforward asymptotic expansion led to