91Ó°ÊÓ

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

Use the Laplace transform and these inverses to solve the given initial-value problem. $$ y^{\prime \prime}-2 y^{\prime}+5 y=0, \quad y(0)=1, y^{\prime}(0)=3 $$

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. $$ f(t)=1+t-\frac{8}{3} \int_{0}^{t}(\tau-t)^{3} f(\tau) d \tau $$

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

Make up two functions \(f_{1}\) and \(f_{2}\) that have the same Laplace transform. Do not think profound thoughts.

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

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

Use the Laplace transform to solve the given integral equation or integrodifferential equation. $$ y^{\prime}(t)=1-\sin t-\int_{0}^{t} y(\tau) d \tau, \quad y(0)=0 $$

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