Chapter 7.6, Problem 15P

This problem deals with a mass in on a spring (with constant k) that receives an impulse $p_0=m{v}_0$ at time t = 0. Show that the initial value problems

$mx\text{'}\text{'}+kx=0;x\left(0\right)=0,x\text{'}\left(0\right)=v_0$ and $mx\text{'}\text{'}+kx=p_0\delta \left(t\right);x\left(0\right)=0,x\text{'}\left(0\right)=0$ have the same solution. Thus the effect of $p_0\delta \left(t\right)$ is, indeed, to impart to the particle an initial momentum $p_0$ .