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where A is an arbitrary constant. Finally, from equation (3.81c), we obtain

which has a periodic solution for only if and so the method
proceeds. Thus the transitional curve in the along which a or
periodic solution exists, is

(b) Case n = 1
This time, although the general procedure is essentially the same, the appearance
of non-trivial periodic solutions at leading order complicates things somewhat.
Equation (3.79) now becomes

and thus we obtain the equations

and so on. The general solution of equation (3.83a) is simply

for arbitrary constants A and B; we may now proceed, collecting all terms pro-
portional to A, and correspondingly to B, but it is far easier (and more usual) to
treat these two sets of terms separately. Thus we select A = 1, B = 0, and A= 0,
B = 1; this will generate two transitional curves: one associated with
and one with which is the usual presentation adopted. Let us choose
then (3.83b) can be written

and a periodic solution for requires (because otherwise there would
be a term proportional to t therefore we obtain