91Ó°ÊÓ

Use the definition of the Laplace transform to find \(\mathscr{Y}\\{f(t)\\}\). \(f(t)= \begin{cases}t, & 0 \leq t<1 \\ 2-t, & t \geq 1\end{cases}\)

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

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

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

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

Use the definition of the Laplace transform to find \(\mathscr{Y}\\{f(t)\\}\). \(f(t)= \begin{cases}0, & 0 \leq t<2 \\ 1, & 2 \leq t<4 \\ 0, & t \geq 4\end{cases}\)

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

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._____

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