91Ó°ÊÓ

Use the Laplace transform to solve the given initial-value problem. $$ y^{\prime}+y=f(t), \quad y(0)=0, \text { where } f(t)=\left\\{\begin{array}{rr} 0, & 0 \leq t<1 \\ 5, & t \geq 1 \end{array}\right. $$

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

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

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

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

Use the Laplace transform to solve the given initial-value problem. \(y^{\prime \prime}+4 y=f(t), \quad y(0)=0, y^{\prime}(0)=-1\), where $$ f(t)=\left\\{\begin{array}{rr} 1, & 0 \leq t<1 \\ 0, & t \geq 1 \end{array}\right. $$

In Problems, use the Laplace transform to solve the given initial-value problem. $$ \begin{gathered} y^{\prime \prime}+4 y=f(t), \quad y(0)=0, y^{\prime}(0)=-1, \text { where } \\\ f(t)=\left\\{\begin{array}{lr} 1, & 0 \leq t<1 \\ 0, & t \geq 1 \end{array}\right. \end{gathered} $$

Use the Laplace transform to solve the given initial-value problem. $$ y^{\prime \prime}-5 y^{\prime}+6 y=9(t-1), \quad y(0)=0, \quad y^{\prime}(0)=1 $$

In Problems, use the Laplace transform to solve the given initial-value problem. $$ y^{\prime \prime}-5 y^{\prime}+6 y=\mathcal{u}(t-1), \quad y(0)=0, y^{\prime}(0)=1 $$

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 $$ f(t)=\left\\{\begin{array}{lr} 0, & 0 \leq t<\pi \\ 1, & \pi \leq t<2 \pi \\ 0, & t \geq 2 \pi \end{array}\right. $$