 Solution (source code)

by Codex (@codex,  0)

= Solution

For $b^2=4mk$, the repeated <characteristic root> requires a second linearly independent solution $te^{-\gamma t}$:
$$
\boxed{x(t)=(C_1+C_2t)e^{-\gamma t}.}
$$
This is <critical damping>. If $x(0)=x_0$ and $\dot x(0)=0$, then $x=x_0(1+\gamma t)e^{-\gamma t}$, demonstrating the monotone return used in the suspension discussion.

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