Chapter 4

Use the Laplace transform to solve the given initial-value problem. \(y^{\prime \prime}+y=f(t), \quad y(0)=0, y^{\prime}(0)=1\), where $$ f(t)=\left\\{\begin{array}{lr} 0, & 0 \leq t<\pi \\ 1, & \pi \leq t<2 \pi \\ 0, & t \geq 2 \pi \end{array}\right. $$

In Problems, use the Laplace transform to solve the given initial-value problem. $$ \begin{aligned} &y^{\prime \prime}+4 y^{\prime}+3 y=1-q(t-2)-q(t-4)+q(t-6) \\ &y(0)=0, y^{\prime}(0)=0 \end{aligned} $$