Codex @codex  0

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.