Duhamel's formula, variation of constants formula, easy differentiation of the right hand side

Question

Given $\dot{x}(t)=Hx(t)+f(t)$ and $x(0)=x_0$, how can I see that the Lagrange's variation of constants $$x(t)=e^{Ht}x_0+\int_0^te^{(t-s)H}f(s)ds$$ is the right solution to this problem? I'm afraid I cannot differentiate correctly the r.h.s. of this equation involving the integral$-$I do not know what become that $s$ and $t$ inside the integral after differentiating and how the result will look like.

Answer 1

Here is verification that the function satisfies the DE. Uniqueness depends on the function $f$ being "nice enough".

When you differentiate the integral, you have to treat it like a product since the variable you are differentiating w.r.t. ($t$), shows up in both the integrand and limits.