 Solution (source code)

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= Solution
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The <convolution theorem for Laplace transforms> states
$$
\boxed{\bar x(p)=\bar y(p)\bar z(p)}
$$
when $x(t)=\int_0^ty(s)z(t-s)\,ds$ and the transforms converge appropriately. One proves it by changing variables $t=s+r$ in the double integral over $0\leq s\leq t<\infty$, which becomes the product domain $s,r\geq0$. Absolute convergence justifies the interchange.

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