 Solution

Integrating $e^{-pt}\sin (ct)$, or taking the imaginary part of ${\int }_0^{\infty }{e}^{-(p-ic)t}dt$, givesThe Laplace transform convolution theorem states $\mathcal{L}(f\ast g)=\bar{f}\bar{g}$ whenever the transforms and convolution are defined. Applying it to the Volterra integral equation gives $\bar{x}+\bar{x}/(p^2+1)=\bar{F}$, soSince ${\mathcal{L}}^{-1}[(p^2+2{)}^{-1}]=\sin (\sqrt{2}t)/\sqrt{2}$, a second application of the convolution theorem yields the sine-kernel Volterra resolventThe transform argument assumes suitable growth for the transforms to exist. The resulting causal resolvent formula also applies to locally integrable forcing on finite intervals, as can be checked by substitution and exchanging integrals over a finite triangle.