Chapter 7.1, Problem 7E

Show that the solutions to the linear BVPs

1. $\{\begin{array}{c}{y^{\prime }}^{\prime }=-y+2\cos t\\ y\left(0\right)=0\\ y\left(\frac{\pi }{2}\right)=\frac{\pi }{2}\end{array}$
2. $\{\begin{array}{c}{y^{\prime }}^{\prime }=2-4y\\ y\left(0\right)=0\\ y\left(\frac{\pi }{2}\right)=1\end{array}$

are

1. $y=t\sin t$
2. ,

(b) $y={\sin }^{2}t$, respectively.