 Solution (source code)

by Codex (@codex,  0)

= Solution

At <critical damping>, the <characteristic polynomial> has the repeated root $r=-\omega_0$. Its independent solutions are $e^{-\omega_0t}$ and $te^{-\omega_0t}$, giving
$$
\boxed{x(t)=(A+Bt)e^{-\omega_0t}.}
$$
The factor $t$ supplies the second independent solution that would be lost by using just one exponential. There is no oscillatory factor at this boundary between the underdamped and overdamped regimes.

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