91Ó°ÊÓ

In Problems 1-12 use the Laplace transform to solve the given system of differential equations. $$ \begin{aligned} &\frac{d^{2} x}{d t^{2}}+\frac{d^{2} y}{d t^{2}}=t^{2} \\ &\frac{d^{2} x}{d t^{2}}-\frac{d^{2} y}{d t^{2}}=4 t \\ &x(0)=8, x^{\prime}(0)=0 \\ &y(0)=0, y^{\prime}(0)=0 \end{aligned} $$

Use the Laplace transform to solve the given differential equation subject to the indicated initial conditions. $$ y^{\prime \prime}+4 y^{\prime}+5 y=\delta(t-2 \pi), \quad y(0)=0, y^{\prime}(0)=0 $$

Use the Laplace transform to solve the given differential equation subject to the indicated initial conditions. $$ y^{\prime \prime}-2 y^{\prime}+5 y=1+t, \quad y(0)=0, \quad y^{\prime}(0)=4 $$

Fill in the blanks or answer true/false. \(\mathscr{L}\left\\{e^{-\mathrm{M}} \sin 2 t\right\\}=\)_____

Find the given inverse transform. \(\mathscr{L}^{-1}\left\\{\frac{1}{5 s-2}\right\\}\)

In Problems 1-12 use the Laplace transform to solve the given system of differential equations. $$ \begin{aligned} &\frac{d x}{d t}-4 x+\frac{d^{3} y}{d t^{3}}=6 \sin t \\ &\frac{d x}{d t}+2 x-2 \frac{d^{3} y}{d t^{3}}=0 \\ &x(0)=0, y(0)=0 \\ &y^{\prime}(0)=0, y^{N}(0)=0 \end{aligned} $$

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

Find the given inverse transform. \(\mathscr{L}^{-1}\left\\{\frac{5}{5^{2}+49}\right\\}\)

Fill in the blanks or answer true/false. \(\mathscr{L}\\{t \sin 2 t\\}=\)_____

Use the Laplace transform to solve the given differential equation subject to the indicated initial conditions. $$ y^{\prime \prime}+4 y^{\prime}+13 y=8(t-\pi)+\delta(t-3 \pi), \quad y(0)=1, y^{\prime}(0)=0 $$