Show that the response of an underdamped, second-order system to an impulsive input \(\delta(t-a), a=\) const \(>0\), and zero initial conditions, is described by

\[\left[\frac{1}{\omega_{d}} e^{-\zeta \omega_{n}(t-a)} \sin \omega_{d}(t-a)\right] u(t-a)\]

where \(u(t)\) is the unit-step function.