= Solution
The moving equilibrium produces a <forced harmonic oscillator>:
$$
\ddot x+\gamma\dot x+\omega_0^2x=\omega_0^2d e^{-\alpha t}.
$$
A particular solution is $F e^{-\alpha t}$, where $F=\omega_0^2d/(\alpha^2-\gamma\alpha+\omega_0^2)$. 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 gives
$$
\boxed{x(t)=F e^{-\alpha t}+(d-F)e^{-\gamma t/2}
\left[\cos\omega_dt+\frac{\gamma/2-\alpha}{\omega_d}\sin\omega_dt\right].}
$$
For $\alpha=\gamma$, $F=d$, so the entire oscillatory term vanishes:
$$
\boxed{x(t)=d e^{-\gamma t}=x_0(t).}
$$
The body is <tracking an exponentially moving equilibrium without oscillation>. There is no spring extension in this special motion: <linear drag> alone supplies $m\ddot x=\gamma^2md e^{-\gamma t}$. An underdamped homogeneous equation does not require every forced solution to contain an oscillatory transient.
\Image[/past-exam-of-the-mathematical-methods-i-course-of-the-university-of-oxford/2016-resit-damped-equilibrium-tracking.png]
{title=Free damped oscillations and exact tracking of an exponentially moving equilibrium}
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