Exercise 29

Given the equation $x' + a x = q(t)$, where $a$ is a constant and $q(t)$ is a continuous function defined on an interval $I$, show that $$x(t) = Ce^{-at} + e^{-at} \int_{t_0}^t e^{as}q(s) \, ds$$ is a solution of this equation, where $c$ is any constant and $t_0 \in I$.