Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 336 / 2 / a

Introduce independent fast and slow timesand seek $u=u_0+ϵu_1+\cdots$. ThenAt leading order,so writeAt order $ϵ$,The solvability condition in the method of multiple scales removes the resonant sine and cosine components. Averaging over one fast period gives the amplitude-phase equations for a weakly perturbed oscillatorTherefore $u\sim R(T)\cos [t+\varphi (T)]$ without the secular growth that a single-time expansion would produce.

At leading order,soThe required period averages areHenceThe initial data give $R(0)=1$ and $\varphi (0)=0$, soThus the perturbation produces a slow frequency shift but no leading amplitude decay:

NowThereforewhereas $R{\varphi }^{\prime }=0$ because $⟨{\cos }^3\theta \sin \theta ⟩=0$. Integrating with $R(0)=1$ givesand henceThe leading uniformly slow-decaying oscillation is