 Solution

The moving equilibrium produces a forced harmonic oscillator:A particular solution is $F{e}^{-\alpha t}$, where $F={\omega }_0^{2}d/({\alpha }^2-\gamma \alpha +{\omega }_0^2)$. The denominator equals $(\alpha -\gamma /2{)}^2+{\omega }_d^2>0$, so it cannot vanish for real $\alpha$ under the given underdamping condition. Imposing both initial conditions givesFor $\alpha =\gamma$, $F=d$, so the entire oscillatory term vanishes:The body is tracking an exponentially moving equilibrium without oscillation. There is no spring extension in this special motion: linear drag alone supplies $m\ddot{x}={\gamma }^{2}md{e}^{-\gamma t}$. An underdamped homogeneous equation does not require every forced solution to contain an oscillatory transient.