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216 5. Some worked examples arising from physical problems

and so, from (5.51), we obtain the equation

which can be integrated directly:

where is an arbitrary constant. This solution is to match to the asymptotic expan-
sion for and, in particular, to the first term, (5.49). Further, we are seek-
ing the condition(s) that ensure the existence of a periodic solution, so we also
impose

In principle, we are able to match (5.49) and (5.52) for arbitrary but the
result is not particularly useful; further, the periodicity condition implies that, for small
then must be small. Thus we again invoke and from (5.52) we obtain
(for

which matches with (5.50f) if we choose

Indeed, this match is valid for The periodicity requirement, (5.53),
now becomes and thus from (5.52) (with and (5.50a) we
obtain

Thus (5.54) and (5.55) imply that we must have

where and so

i.e. for the given we have