91Ó°ÊÓ

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

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

Use the Laplace transform to solve the given initial-value problem. \(y^{\prime}+2 y=f(t), \quad y(0)=0,\) 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. \(y^{\prime \prime}+4 y=f(t), \quad y(0)=0, \quad y^{\prime}(0)=-1,\) where \\[ f(t)=\left\\{\begin{array}{lr} 1, & 0 \leq t<1 \\ 0, & t \geq 1 \end{array}\right. \\]

The Laplace transform \(\mathscr{L}\left\\{e^{-r^{2}}\right\\}\) exists, but without finding it solve the initial-value problem \(y^{\prime \prime}+y=e^{-t^{2}}, y(0)=0\) \(y^{\prime}(0)=0\)

Solve the integral equation $$f(t)=e^{t}+e^{t} \int_{0}^{t} e^{-\tau} f(\tau) d \tau$$

Use the Laplace transform to solve the given initial-value problem. $$y^{\prime \prime}+4 y=\sin t \mathscr{U}(t-2 \pi), \quad y(0)=1, \quad y^{\prime}(0)=0$$

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

Use the Laplace transform as an aide in evaluating the improper integral \(\int_{0}^{x} t e^{-2 t} \sin 4 t d t\)

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