= Solution
After the force is removed, the <damped harmonic oscillator> evolves freely. There is no impulsive force at zero, so both displacement and velocity are continuous. Matching the values in part (i) gives
$$
\boxed{x(t)=e^{-\gamma t}\left[x_0\cos\nu t+\frac{v_0+\gamma x_0}{\nu}\sin\nu t\right],\qquad t>0.}
$$
Equivalently,
$$
x(t)=\frac{A\omega e^{-\gamma t}}N\left[-2\gamma\cos\nu t+\frac{\Omega^2-\omega^2-2\gamma^2}{\nu}\sin\nu t\right].
$$
This is bounded in positive time, completing the unique globally bounded solution. The formulas also include $\omega=0$, for which the forcing and solution both vanish.
\Image[/past-exam-of-the-mathematical-methods-i-course-of-the-university-of-oxford/2015-force-removal-ringdown.png]
{title=Steady forced motion before switch-off joins continuously to a freely decaying oscillation}
Back to article page
