Chapter 4

Use the Laplace transform to solve the given differential equation subject to the indicated initial conditions. $$ y^{\prime}-3 y=\delta(t-2), \quad y(0)=0 $$

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

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

Use the definition of the Laplace transform to find \(\mathscr{L}\\{f(t)\\}\). $$ f(t)=\left\\{\begin{array}{lr} t, & 0 \leq t<1 \\ 2-t, & t \geq 1 \end{array}\right. $$

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

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

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

In Problems, find either \(F(s)\) or \(f(t)\), as indicated. $$ \mathscr{L}\left\\{t e^{-6 t}\right\\} $$

Use the definition of the Laplace transform to find \(\mathscr{L}\\{f(t)\\}\). $$ f(t)=\left\\{\begin{array}{lr} 0, & 0 \leq t<2 \\ 1, & 2 \leq t<4 \\ 0, & t \geq 4 \end{array}\right. $$

Find either \(F(s)\) or \(f(t)\), as indicated. $$ \mathscr{L}\left\\{t e^{-6 t}\right\\} $$

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

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

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.___

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

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

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

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

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

Fill in the blanks or answer true/false. The function \(f(t)=\left(e^{t}\right)^{10}\) is not of exponential order.___

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

Use the Laplace transform to solve the given differential equation subject to the indicated initial conditions. $$ y^{\prime \prime}+y=\delta(t-\pi / 2)+\delta(t-3 \pi / 2), \quad y(0)=0, y^{\prime}(0)=0 $$

Use the Laplace transform to solve the given system of differential equations. $$ \begin{aligned} &2 \frac{d x}{d t}+\frac{d y}{d t}-2 x=1 \\ &\frac{d x}{d t}+\frac{d y}{d t}-3 x-3 y=2 \\ &x(0)=0, y(0)=0 \end{aligned} $$

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

Fill in the blanks or answer true/false. \(F(s)=s^{2} /\left(s^{2}+4\right)\) is not the Laplace transform of a function that is piecewise continuous and of exponential order.____

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

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. $$ \begin{aligned} &\frac{d x}{d t}+x-\frac{d y}{d t}+y=0 \\ &\frac{d x}{d t}+\frac{d y}{d t}+2 y=0 \\ &x(0)=0, y(0)=1 \end{aligned} $$

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

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

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

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

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. $$ y^{\prime \prime}-2 y^{\prime}=1+\delta(t-2), \quad y(0)=0, y^{\prime}(0)=1 $$

Use the Laplace transform to solve the given system of differential equations. $$ \begin{aligned} &\frac{d^{2} x}{d t^{2}}+\frac{d x}{d t}+\frac{d y}{d t}=0 \\ &\frac{d^{2} y}{d t^{2}}+\frac{d y}{d t}-4 \frac{d x}{d t}=0 \\ &x(0)=1, x^{\prime}(0)=0 \\ &y(0)=-1, y^{\prime}(0)=5 \end{aligned} $$

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

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. $$ \mathscr{L}\left\\{e^{-2 t} \cos 4 t\right\\} $$

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 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 initial-value problem. Use the table of Laplace transforms in Appendix III as needed. $$ y^{\prime}+y=t \sin t, \quad y(0)=0 $$

In Problems, find either \(F(s)\) or \(f(t)\), as indicated. $$ \mathscr{L}\left\\{\left(1-e^{t}+3 e^{-4 t}\right) \cos 5 t\right\\} $$

Fill in the blanks or answer true/false. $$ \mathscr{L}\\{\sin 2 t\\}= $$____

Find either \(F(s)\) or \(f(t)\), as indicated. $$ \mathscr{L}\left\\{\left(1-e^{t}+3 e^{-4 t}\right) \cos 5 t\right\\} $$

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

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

Use the Laplace transform to solve the given initial-value problem. Use the table of Laplace transforms in Appendix III as needed. $$ y^{\prime}-y=t e^{t} \sin t, \quad y(0)=0 $$