 Solution

The result of part (i) vanishes identically on the open interval $t<0$, so its first and second derivatives also vanish there. Substituting into the equation of motion immediately givesThus the force has no negative-time prehistory, consistent with the causal Green function of a damped oscillator. In fact, direct inversion or the positive-time substitution above identifies $f(t)=-e^{-2t}H(t)$; the given numerator $+i$ corresponds to a negative exponential force with this Fourier convention. Here $H$ is the Heaviside step function; its value exactly at the jump does not affect the transform.