Chapter 6.6, Problem 22P

(a)

To determine

The solution of the integral equation $\varphi \left(t\right)+{\displaystyle \underset{0}{\overset{t}{\int }}\left(t-\xi \right)}\varphi \left(\xi \right)d\xi =\sin 2t$, using Laplace transform.

(b)

To determine

To show: That the function $\varphi \left(t\right)$ satisfies the differential equation ${\varphi }^{″}\left(t\right)+\varphi \left(t\right)=-4\sin 2t$ and the initial conditions $\varphi \left(0\right)=0\text{ and }{\varphi }^{\prime }\left(0\right)=2$.

(c)

To determine

To solve: The initial value problem in the sub part (b) and to check whether the obtained solution is same as the one obtained in sub part (a).