Tracking an exponentially moving equilibrium without oscillation ($x_0=d{e}^{-\gamma t},x=x_0$)

For $\ddot{x}+\gamma \dot{x}+{\omega }_0^2(x-x_0)=0$, the particular path $x=x_0$ is a solution precisely when ${\ddot{x}}_0+\gamma {\dot{x}}_0=0$. An exponentially moving equilibrium $x_0=d{e}^{-\gamma t}$ satisfies this condition. If the initial position and velocity match it, uniqueness removes all homogeneous oscillations even when the natural system is underdamped. The motion is sustained dynamically by linear drag, with zero instantaneous spring extension.