= Solution
The <small-angle approximation> replaces $\sin\theta$ by $\theta$, giving the <damped harmonic oscillator> equation $\ddot\theta+c\dot\theta+\theta=0$. Its <characteristic roots> satisfy $r^2+cr+1=0$.
For \b[$0<c<2$], put $\nu=\sqrt{1-c^2/4}$; the motion is underdamped:
$$
\boxed{\theta(t)\simeq e^{-ct/2}[A\cos(\nu t)+B\sin(\nu t)].}
$$
It oscillates with decaying amplitude. For \b[$c=2$], the repeated root gives <critical damping>:
$$
\boxed{\theta(t)\simeq(A+Bt)e^{-t}.}
$$
For \b[$c>2$], the motion is overdamped:
$$
\boxed{\theta(t)\simeq A e^{(-c+\sqrt{c^2-4})t/2}+B e^{(-c-\sqrt{c^2-4})t/2}.}
$$
Both real exponents are negative, so there is no oscillatory factor. The two constants are arbitrary in each case; these approximations require the displacement to remain small.
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