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

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

Set $X=ϵx$. The equation becomesSince $k$ is positive for every $X\ge 0$, there is no classical turning point. The leading WKB approximation for a slowly varying oscillator isThe condition $y(0)=0$ selects a sine, and $y_x(0)=1$ fixes its coefficient. Sincethe result isThe WKB validity measureis uniformly $O(ϵ)$ and tends to zero at both ends of $X\ge 0$. With no turning point, the approximation is therefore uniformly valid on the entire stated half-line.