I'm trying to solve a (linear, homogeneous) ODE (and find the function f(t)) which includes the convolution \begin{equation} \begin{aligned} (G*f)(t) := \int_{-\infty}^tG(t - u)f(u)du + \int_{t}^\infty G(u - t)f(u)du \ . \end{aligned} \end{equation}
Here $G(t)$ is the Green's function that satisfies \begin{equation} \begin{aligned} \frac{d^2G}{dt^2}(t - u) + G(t - u) = \delta(t - u) \ , \end{aligned} \end{equation}
where $\delta(t)$ is the Dirac delta distribution. I thought I might try to solve my ODE with a Laplace transform \begin{equation} \begin{aligned} \mathcal{L}\{f\}(s) := \int_0^\infty dte^{-st}f(t) \ . \end{aligned} \end{equation}
(The two-sided Laplace transform is integrated from "$-\infty$" to $\infty$.)
However I'm unsure whether the Laplace transform factorizes my convolution \begin{equation} \begin{aligned} \mathcal{L}\{f*g\}(s) = \mathcal{L}\{f\}\mathcal{L}\{g\} \ ? \end{aligned} \end{equation}