91Ó°ÊÓ

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

Use the Laplace transform to solve the given initial-value problem. $$2 y^{\prime \prime \prime}+3 y^{\prime \prime}-3 y^{\prime}-2 y=e^{-t}, \quad y(0)=0, \quad y^{\prime}(0)=0, \quad y^{\prime \prime}(0)=1$$

Find either \(F(s)\) or \(f(t),\) as indicated. $$\mathscr{L}^{-1}\left\\{\frac{e^{-2 s}}{s^{3}}\right\\}$$

Use the Laplace transform to solve the given integral equation or integrodifferential equation. $$f(t)=t e^{t}+\int_{0}^{t} \tau f(t-\tau) d \tau$$

Use the Laplace transform to solve the given initial-value problem. $$y^{\prime \prime \prime}+2 y^{\prime \prime}-y^{\prime}-2 y=\sin 3 t, \quad y(0)=0, \quad y^{\prime}(0)=0, \quad y^{\prime \prime}(0)=1$$

Find either \(F(s)\) or \(f(t),\) as indicated. $$\mathscr{L}^{-1}\left\\{\frac{\left(1+e^{-2 s}\right)^{2}}{s+2}\right\\}$$

Use the Laplace transform to solve the given integral equation or integrodifferential equation. $$f(t)+2 \int_{0}^{t} f(\tau) \cos (t-\tau) d \tau=4 e^{-t}+\sin t$$

Use the Laplace transform to solve the given integral equation or integrodifferential equation. $$f(t)+\int_{0}^{t} f(\tau) d \tau=1$$

Find either \(F(s)\) or \(f(t),\) as indicated. $$\mathscr{L}^{-1}\left\\{\frac{s e^{-\pi s / 2}}{s^{2}+4}\right\\}$$

Use the Laplace transform to solve the given integral equation or integrodifferential equation. $$f(t)=\cos t+\int_{0}^{t} e^{-\tau} f(t-\tau) d \tau$$