= Solution
Let $r_1,r_2$ be the roots of $\alpha r^2+\beta r+\gamma=0$. The <causal Green function> is $G(t,\tau)=H(t-\tau)g(t-\tau)$, where
$$
g(s)=\frac{e^{r_1s}-e^{r_2s}}{\alpha(r_1-r_2)}
$$
for distinct roots, and $g(s)=se^{rs}/\alpha$ for a repeated root. These formulas enforce $g(0)=0$ and $\alpha g'(0)=1$, hence the required delta jump.
For the oscillator,
$$
y(t)=\int_0^t\frac{\sin(\omega(t-\tau))}{\omega}\sin(\lambda\tau)\,d\tau.
$$
If $\lambda\ne\omega$, this is
$$
y(t)=\frac{\sin(\lambda t)-(\lambda/\omega)\sin(\omega t)}{\omega^2-\lambda^2}.
$$
At resonance $\lambda=\omega$, the limiting expression is
$$
\boxed{y(t)=\frac{\sin(\omega t)-\omega t\cos(\omega t)}{2\omega^2}.}
$$
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