91Ó°ÊÓ

In each of the following exercises, use the Laplace transform to find the solution of the given linear system that satisfies the given initial conditions. \(x^{\prime}+y=3 e^{2 t}\) \(y^{\prime}+x=0\) \(x(0)=2, y(0)=0\)

Use the Laplace transforms to solve each of the initial-value. \(y^{\prime}-y=e^{3 t}\) \(y(0)=2\)

Find \(\mathcal{L}\\{f(t)\\}\) for each of the functions \(f\) defined. $$ f(t)=\left\\{\begin{array}{ll} 0, & 06 \end{array}\right. $$

Use the Laplace transforms to solve each of the initial-value. \(y^{\prime}+y=2 \sin t\) \(y(0)=-1\)

In each of the following exercises, use the Laplace transform to find the solution of the given linear system that satisfies the given initial conditions. \(x^{\prime}-2 y=0\) \(y^{\prime}+x-3 y=2\) \(x(0)=3, y(0)=0\)

Find \(\mathcal{L}\\{f(t)\\}\) for each of the functions \(f\) defined. $$ f(t)=\left\\{\begin{aligned} 0, & 010 \end{aligned}\right. $$

In each of the following exercises, use the Laplace transform to find the solution of the given linear system that satisfies the given initial conditions. \(x^{\prime}-5 x+2 y=3 e^{4 t}\) \(y^{\prime}-4 x+y=0\) \(x(0)=3, y(0)=0\)

Use Table \(9.1\) to find \(\mathcal{I}^{-1}\\{F(s)\\}\) for each of the functions \(F\) defined \(F(s)=\frac{2}{s^{2}+9}\)

Find \(\mathcal{L}\\{f(t)\\}\) for each of the functions \(f\) defined. $$ f(t)=\left\\{\begin{array}{ll} 4, & 06 \end{array}\right. $$

Use the Laplace transforms to solve each of the initial-value. \(y^{\prime}+4 y=6 e^{-t}\) \(y(0)=5\)