Damped harmonic oscillator

Example

Let $m > 0$, $k > 0$ and $c > 0$ be constants. The damped harmonic oscillator is the equation $$m\, x'' + c\, x' + k\, x = 0, \quad \text{that is} \quad x'' + 2 \zeta \omega_0\, x' + \omega_0^2\, x = 0,$$ with $\omega_0 = \sqrt{k/m}$ and damping ratio $\zeta = c / \left(2 \sqrt{m k}\right)$. The characteristic roots are $r = \omega_0 \left(-\zeta \pm \sqrt{\zeta^2 - 1}\right)$, and there are three cases. If $\zeta > 1$ (overdamped), the roots are real, distinct and negative, and $x = c_1 e^{r_1 t} + c_2 e^{r_2 t}$. If $\zeta = 1$ (critically damped), $r = -\omega_0$ is a double root, and $x = (c_1 + c_2 t) e^{-\omega_0 t}$. If $0 < \zeta < 1$ (underdamped), $x = e^{-\zeta \omega_0 t} \left(c_1 \cos \omega_1 t + c_2 \sin \omega_1 t\right)$ with $\omega_1 = \omega_0 \sqrt{1 - \zeta^2}$.