91Ó°ÊÓ

Use the Laplace transform to solve the given differential equation subject to the indicated initial conditions. $$ y^{\prime \prime}+y=\delta(t-2 \pi)+\delta(t-4 \pi), \quad y(0)=1, y^{\prime}(0)=0 $$

Use the Laplace transform to solve the given system of differential equations. $$ \begin{aligned} &\frac{d^{2} x}{d t^{2}}+x-y=0 \\ &\frac{d^{2} x}{d t^{2}}+y-x=0 \\ &x(0)=0, x^{\prime}(0)=-2, \\ &y(0)=0, y^{\prime}(0)=1 \end{aligned} $$

Use Theorem to evaluate the given Laplace transform. $$ \mathscr{L}\left\\{t e^{2 t} \sin 6 t\right\\} $$

In Problems, find either \(F(s)\) or \(f(t)\), as indicated. $$ \mathscr{L}\left\\{e^{t} \sin 3 t\right\\} $$

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

Use the Laplace transform to solve the given differential equation subject to the indicated initial conditions. $$ y^{\prime \prime}+2 y^{\prime}=\delta(t-1), \quad y(0)=0, y^{\prime}(0)=1 $$

Fill in the blanks or answer true/false. $$ \mathscr{L}\left\\{t e^{-7 t}\right\\}= $$____

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

In Problems, find either \(F(s)\) or \(f(t)\), as indicated. $$ \mathscr{L}\left\\{e^{-2 t} \cos 4 t\right\\} $$

Use the Laplace transform to solve the given differential equation subject to the indicated initial conditions. $$ y^{\prime \prime}-2 y^{\prime}=1+\delta(t-2), \quad y(0)=0, y^{\prime}(0)=1 $$