Forced oscillation and resonance

Example

Let $\omega_0 > 0$, $\zeta \ge 0$, $F > 0$ and $\omega > 0$ be constants. The forced oscillator is $$x'' + 2 \zeta \omega_0\, x' + \omega_0^2\, x = F \cos(\omega t).$$ If $\zeta > 0$, it has the particular solution $x_p(t) = a(\omega) \cos(\omega t - \delta)$ for a phase $\delta$, with amplitude $$a(\omega) = \frac{F}{\sqrt{\left(\omega_0^2 - \omega^2\right)^2 + \left(2 \zeta \omega_0 \omega\right)^2}}.$$ Each solution is $x_p$ plus a solution of the damped equation, which tends to $0$. So $x_p$ is the steady state. If $\zeta = 0$ and $\omega = \omega_0$, a particular solution is $x_p(t) = \frac{F}{2 \omega_0}\, t \sin(\omega_0 t)$, which is unbounded. This is resonance.