= Solution
The <method of multiple scales> introduces independent variables for the fast oscillation and its slow modulation, for example $t_0=t$, $T=\epsilon t$, and, when needed, $T_2=\epsilon^2t$. The derivative becomes $d/dt=\partial_{t_0}+\epsilon\partial_T+\epsilon^2\partial_{T_2}+\cdots$. One expands the solution in $\epsilon$ while allowing its leading <amplitudes> and phases to depend on the slow times.
A <solvability condition in the method of multiple scales> removes resonant forcing of each fast homogeneous mode. Otherwise a correction contains a <secular term>, such as $\epsilon t\sin t$, and an ordinary <asymptotic expansion> fails on long times. The resulting slow <amplitude> equations incorporate that accumulated effect in the leading solution. One must still check that the <amplitudes> remain within the assumed weak-coupling regime and that omitted slower effects have not accumulated.
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