Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 336 / 2 / a / Solution

Introduce $T=ϵt$ and write $x_0=A(T)\cos [t+\theta (T)]$. Averagingover one fast period gives $A{A}_T=-3A^4/8$ and ${\theta }_T=0$. The initial data therefore give the method of multiple scales resultthrough $t=O({ϵ}^{-1})$.

For the replacement damping, $\sin (\dot{x})\dot{x}={\dot{x}}^2-{\dot{x}}^4/6+\cdots$ is even in $\dot{x}$. Every term has zero resonant projection onto the fundamental over a complete orbit, so the $O(ϵ)$ slow amplitude and phase equations vanish. Thusthrough $t=O({ϵ}^{-1})$: there is no first-order secular damping, although bounded mean and higher-harmonic corrections occur.