= Tracking an exponentially moving equilibrium without oscillation
{title2=$x_0=d e^{-\gamma t},\quad x=x_0$}
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.
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