Chapter 3.7, Problem 10P

To determine

To prove: The solution of the initial value problem $m{u}^{″}+\gamma u^{\prime }+ku=0$, $u\left(t_0\right)=u_0$, $u^{\prime }\left(t_0\right)=u_0{}^{\prime }$ can be expressed as the sum $u=v+w$, where v satisfies the initial conditions $v\left(t_0\right)=u_0$, $v^{\prime }\left(t_0\right)=0$ and w satisfies the conditions $w\left(t_0\right)=0$, $w^{\prime }\left(t_0\right)=u_0{}^{\prime }$.