 Laplace-transform solution of a constant-coefficient vector ODE

For a constant matrix $A$ and a forcing vector-valued function $g$, considerWhen the component Laplace transforms converge and the Laplace transform of a derivative applies, setting $Y=\mathcal{L}y$ and $G=\mathcal{L}g$ givesChoose $s$ in the convergence half-plane and outside the spectrum of $A$. The resolvent $(sI-A{)}^{-1}$ is the Laplace transform of the matrix exponential $e^{At}$ for sufficiently large real part. Inverting gives the variation-of-constants formulaThe matrix acts on the forcing from the left, so there is no implicit commutation of a time-dependent forcing with $A$.