= Solution
Take the causal solution, extending the input and output by zero before the device starts at $t=0$. The two zero <initial conditions> ensure that this extension adds no impulsive terms to the first two <distributional derivatives>. The <Fourier transform of a derivative> is multiplication by $i\omega$, so
$$
\left[(p+i\omega)^2+q^2\right]\widetilde\phi(\omega)=\widetilde I(\omega).
$$
Factoring the quadratic and taking <partial fraction decomposition> yields
$$
\boxed{\widetilde\phi=\widetilde R\widetilde I,\qquad
\widetilde R(\omega)=\frac1{2iq}\left(\frac1{p+i\omega-iq}-\frac1{p+i\omega+iq}\right).}
$$
The inverse <Fourier transform> follows directly by integrating decaying exponentials:
$$
\boxed{R(t)=H(t)\frac{e^{-pt}\sin(qt)}q,}
$$
where $H$ is the <Heaviside step function>. This <damped-oscillator impulse response> has $R(0^+)=0$ and $R'(0^+)=1$, as required for a unit impulse.
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