 Solution

Use the causal Green function: it vanishes before the forcing time $t^{\prime }$. For $t>t^{\prime }$ it solves the homogeneous equation, is continuous at $t=t^{\prime }$ with value zero, and has first-derivative jump one. These conditions giveIndeed the right derivative at the forcing time is one and the left derivative is zero, so $({\partial }_t^2-k^2)G=\delta (t-t^{\prime })$ distributionally. If $k=0$, use the continuous limit $G(t,t^{\prime })=(t-t^{\prime })$ on its causal support.

For $k\ne 0$, differentiating the integral solution givesBoth initial values vanish and $\ddot{x}-k^{2}x=f$. This verifies the requested representation and its sign. The $k=0$ formula similarly gives $\ddot{x}=f$.