Chapter 3.6, Problem 23P

(a)

To determine

To prove: The solution of the differential equation $y^{″}+y=g\left(t\right)$ with the initial conditions $y\left(t_0\right)=0$ and $y^{\prime }\left(t_0\right)=0$ is $Y\left(t\right)={\displaystyle {\int }_{t_0}^t\sin \left(t-s\right)g\left(s\right)}ds$.

(b)

To determine

To solve: The differential equation $y^{″}+y=g\left(t\right)$ with the initial conditions $y\left(0\right)=y_0$ and $y^{\prime }\left(0\right)=y_0{}^{\prime }$.