91Ó°ÊÓ

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

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

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

Fill in the blanks or answer true/false. If \(f\) is not piecewise continuous on \([0, \infty)\), then \(\mathscr{L}\\{f(t)\\}\) will not exist.___

Use the Laplace transform to solve the given system of differential equations. $$ \begin{aligned} &\frac{d x}{d t}=x-2 y \\ &\frac{d y}{d t}=5 x-y \\ &x(0)=-1, y(0)=2 \end{aligned} $$

Use the Laplace transform to solve the given system of differential equations. $$ \begin{aligned} &\frac{d x}{d t}+3 x+\frac{d y}{d t}=1 \\ &\frac{d x}{d t}-x+\frac{d y}{d t}-y=e^{t} \\ &x(0)=0, y(0)=0 \end{aligned} $$

In Problems, find either \(F(s)\) or \(f(t)\), as indicated. $$ \mathscr{L}\left\\{t^{10} e^{-7 t}\right\\} $$

Find either \(F(s)\) or \(f(t)\), as indicated. $$ \mathscr{L}\left\\{t^{10} e^{-7 t}\right\\} $$

Use Theorem to evaluate the given Laplace transform. $$ \mathscr{L}\\{t \sinh 3 t\\} $$

Use the Laplace transform to solve the given differential equation subject to the indicated initial conditions. $$ y^{\prime \prime}+16 y=\delta(t-2 \pi), \quad y(0)=0, y^{\prime}(0)=0 $$