Past exam of the mathematical methods I course of the University of Oxford / 2017 resit / 7 / b

For the damped harmonic oscillator, the characteristic polynomial is $r^2+2\gamma r+{\omega }_0^2$. When $0\le \gamma <{\omega }_0$, its roots are $-\gamma \pm i\Omega$, with $\Omega =\sqrt{{\omega }_0^2-{\gamma }^2}$. ThusHere $A$ and $B$ are arbitrary real constants. The frequency is $\Omega$, and the amplitude envelope decays with time constant $1/\gamma$ when $\gamma >0$; the zero-damping limit recovers simple harmonic motion.

At critical damping, the characteristic polynomial has the repeated root $r=-{\omega }_0$. Its independent solutions are $e^{-{\omega }_{0}t}$ and $t{e}^{-{\omega }_{0}t}$, givingThe factor $t$ supplies the second independent solution that would be lost by using just one exponential. There is no oscillatory factor at this boundary between the underdamped and overdamped regimes.