Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 79 / 2 / a / Solution

At fixed $t$, expand the exact damped harmonic oscillator solution using $e^{-ϵt}=1-ϵt+O({ϵ}^2)$, $\omega =1+O({ϵ}^2)$ and $1/\omega =1+O({ϵ}^2)$. Its Poincaré expansion through first order isThe correction $-ϵt\sin t$ is a secular term. When $ϵt$ is order one it is not small, and the truncated amplitude eventually changes sign and grows in magnitude, instead of reproducing positive damping. This loss of ordering is why a fixed-time regular perturbation is unsuitable on the damping timescale.