Answer
(a) $y(t)=ae^{-ct}sin(\omega t)$
(b) $y(t)=ae^{-ct}cos(\omega t)$
Work Step by Step
General forms for damped harmonic motion $y(t)=ke^{-ct}sin(\omega t)$ and $y(t)=ke^{-ct}cos(\omega t)$
(a) $y(0)=0$ choose sine, we have $y(t)=ae^{-ct}sin(\omega t)$
(b) $y(0)=a$ choose cosine, we have $y(t)=ae^{-ct}cos(\omega t)$
