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Figure 13. Sketch of the solution (5.49) for small

(Most of these results require the use of asymptotic estimates of
for various t, as see e.g. Olver, 1974, or Copson, 1967; cf. Q1.16.) This
(approximate) solution is sketched in figure 13. Clearly and so this
first term is not periodic; however, for the solution is exponentially small
(as and if this becomes as small as then the term omitted in (5.48)
cannot be ignored. (This observation also follows when we find the next term in the
asymptotic expansion and seek a breakdown; this is a rather tiresome process here, so
we treat the problem as one of rescaling the differential equation, as discussed in
For although the precise domain is not yet known, we write
and then (5.47) becomes

We seek a solution