Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 82 / 2 / 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 $ϵt\sin (\omega t)$, which cease to be small on the time scale of interest. Introduce independent fast and slow times, for example $t$ and $T=ϵt$, 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.