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162 4. The method of multiple scales

leaving

Thus, at this stage, we have the asymptotic solution

but and are, as yet, unknown. Before we proceed to examine the
equation for it is instructive to note, in (4.13), that the oscillatory component of
the solution depends on

Thus, at this order, vanishes identically and so we may just as well set this
is the usual simplification that is adopted in these problems. The reason for this redun-
dancy readily becomes clear: in the definition of the fast scale, the term could be
written and then subsumed into the general of the solution. Thus
to set from the outset is permitted.
The equation for from (4.7c) and with becomes

where to make plain all the terms (sin T, cos T) which generate
secular behaviour, we further write this as

Thus is periodic in T if

with initial conditions (from (4.48))

and so