Chapter 10.3, Problem 15P

To determine

The formal solution of the initial value problem $y^{″}+{\omega }^{2}y=f\left(t\right)$, with the initial conditions $y\left(0\right)=0,y^{\prime }\left(0\right)=0$ for the function $f\left(t\right)=\{\begin{array}{l}1,0<t<\pi \\ 0,t=0,\pi ,2\pi \\ -1,\pi <t<2\pi \end{array}$.