Chapter 3.6, Problem 21P

To determine

To prove: The solution of the initial value problem $L\left[y\right]=y^{″}+p\left(t\right)y^{\prime }+q\left(t\right)y=g\left(t\right)$, $y\left(t_0\right)=y_0$, $y^{\prime }\left(t_0\right)=y_0{}^{\prime }$ can be written as $y=u\left(t\right)+v\left(t\right)$ where u and v are solutions of the two initial value problem $L\left[u\right]=0$,  $u\left(t_0\right)=y_0$,  $u^{\prime }\left(t_0\right)=y_0{}^{\prime }$ and $L\left[v\right]=g\left(t\right)$,  $v\left(t_0\right)=0$,  $v^{\prime }\left(t_0\right)=0$, respectively.