Past exam of the mathematical methods I course of the University of Oxford / 2016 resit / 11 / a / Solution

Assume $k>0$, $m>0$ and $\gamma \ge 0$. The spring constant $k$ measures restoring force per displacement from the equilibrium position $x_0$. The term $-\gamma mv$ is linear drag, with damping rate $\gamma$ in inverse-time units. Writing ${\omega }_0=\sqrt{k/m}$, the equation isFor small positive damping it gives underdamped motion: oscillations about $x_0$ with a decaying amplitude. If $\gamma =0$, the motion is undamped simple harmonic motion.