Consider the differential equation for $x(t)$ in $t>0$

$$\ddot{x}-k^{2} x=f(t),$$

subject to boundary conditions $x(0)=0$, and $\dot{x}(0)=0$. Find the Green function $G\left(t, t^{\prime}\right)$ such that the solution for $x(t)$ is given by

$$x(t)=\int_{0}^{t} G\left(t, t^{\prime}\right) f\left(t^{\prime}\right) d t^{\prime} .$$