= Solution
Set the drive to zero and try $x=e^{st}$. The <characteristic equation> is
$$
ms^2+bs+k=0,\qquad s_\pm=-\gamma\pm\sqrt{\gamma^2-\omega_0^2},
\quad\gamma=\frac b{2m},\quad\omega_0^2=\frac km.
$$
For <underdamping>, $\boxed{b^2<4mk}$, the roots are $-\gamma\pm i\omega_d$, with $\omega_d=\sqrt{\omega_0^2-\gamma^2}$. The displacement oscillates with an exponentially decaying envelope:
$$
x=e^{-\gamma t}[C\cos(\omega_dt)+D\sin(\omega_dt)].
$$
For <critical damping>, $\boxed{b^2=4mk}$, there is a repeated root $-\omega_0$, giving $x=(C+Dt)e^{-\omega_0t}$. The motion is nonoscillatory and is the limiting case between oscillatory and overdamped response.
For <overdamping>, $\boxed{b^2>4mk}$, the roots are distinct negative real numbers and $x=C e^{s_+t}+D e^{s_-t}$. Both components decay without sustained oscillation; strong damping gives a particularly slow late-time relaxation. Constants $C,D$ depend on the initial conditions. A nonoscillatory solution can cross equilibrium once for some initial data, so “nonoscillatory” does not mean monotone for every possible initial condition.
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