Initial-velocity memory in the overdamped slow mode

For $m\ddot{x}+\gamma \dot{x}+kx=0$, let $r_s$ and $r_f$ be the slow and fast negative roots. The coefficient of the slow normal mode is exactlyWith $ϵ=km/{\gamma }^2\ll 1$ and ${\tau }_m=m/\gamma$, expansion gives $A_s=x_0+{\tau }_m{v}_0+O(ϵ)(|x_0|+|{\tau }_m{v}_0|)$. Thus the initial velocity can leave a finite displacement after its own fast decay. In particular, $x_0=0$ and $v_0\ne 0$ produce a nonzero slow tail. A first-order overdamped approximation needs an adjusted initial position after the boundary layer; setting it to $x_0$ requires the additional smallness of ${\tau }_m{v}_0$.