= Solution
The <method of multiple scales> applies when a small perturbation acts cumulatively over a long time, as in weakly perturbed <harmonic oscillators> or slowly modulated waves. A regular <asymptotic expansion> can contain <secular terms>, such as $\epsilon t\sin(\omega t)$, which cease to be small on the time scale of interest. Introduce independent fast and slow times, for example $t$ and $T=\epsilon t$, and allow the leading <amplitude> and phase to depend on $T$. The <solvability condition in the method of multiple scales> removes resonant forcing at each order and determines that slow evolution. For the <Mathieu equation>, these <amplitude> equations reveal <parametric resonance> and its instability intervals.
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