Derive Equation 4.19: [ v(t)=frac{1}{omega_{n}^{2}}left[1-e^{-zeta omega_{n} t}left(cos omega_{d} t+frac{zeta omega_{n}}{omega_{d}} sin omega_{d} t ight) ight] ] for

Derive Equation 4.19:

\[ v(t)=\frac{1}{\omega_{n}^{2}}\left[1-e^{-\zeta \omega_{n} t}\left(\cos \omega_{d} t+\frac{\zeta \omega_{n}}{\omega_{d}} \sin \omega_{d} t\right)\right] \]

for \(t \geq 0\).