= 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 gives
$$
\boxed{f(t)=0\quad(t<0).}
$$
Thus 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.
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