Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 316 / 3 / vi / Solution

For $t\gg \tau$, the free term has disappeared. Exactly at $A=g_1$, the resonant contribution tends towith finite eccentricity $|{\nu }_1|\tau$ and a quarter-cycle phase shift. Without damping, exact resonance instead giveswhose secular resonance amplitude grows linearly in the ideal linear theory.