91Ó°ÊÓ

Use the Laplace transforms to solve each of the initial-value. \(y^{\prime \prime}+4 y=8\) \(y(0)=0, \quad y^{\prime}(0)=6\)

Use the Laplace transforms to solve each of the initial-value. \(y^{\prime \prime}+9 y=36 e^{-3 t}\) \(y(0)=2, \quad y^{\prime}(0)=3\)

Find \(\mathcal{L}\\{f(t)\\}\) for each of the functions \(f\) defined. $$ f(t)=\left\\{\begin{array}{ll} 4, & 010 \end{array}\right. $$

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. \(x^{\prime \prime}-3 x^{\prime}+y^{\prime}+2 x-y=0\) \(x^{\prime}+y^{\prime}-2 x+y=0\) \(x(0)=0, y(0)=-1, x^{\prime}(0)=0\)

Use the Laplace transforms to solve each of the initial-value. \(y^{\prime \prime}+6 y^{\prime}+8 y=16\) \(y(0)=0, \quad y^{\prime}(0)=10\)

Use Table \(9.1\) to find \(\mathcal{I}^{-1}\\{F(s)\\}\) for each of the functions \(F\) defined \(F(s)=\frac{s-2}{s^{2}+5 s+6}\)

Find \(\mathcal{L}\\{f(t)\\}\) for each of the functions \(f\) defined. $$ f(t)=\left\\{\begin{array}{ll} 0, & 02 \end{array}\right. $$

Find \(\mathcal{L}\\{f(t)\\}\) for each of the functions \(f\) defined. $$ f(t)=\left\\{\begin{array}{ll} 0, & 04 \end{array}\right. $$

Use formulas \((9.11)\) and \((9.18)\) and Example \(9.4\) to find \(\$\\{f(t)\\}\) if $$ \begin{gathered} 3 f^{\prime \prime}(t)-5 f^{\prime}(t)+7 f(t)=\sin 2 t \\ f(0)=4, \text { and } f^{\prime}(0)=6 \end{gathered} $$

Use the Laplace transforms to solve each of the initial-value. \(2 y^{\prime \prime}+y^{\prime}=5 e^{2 t}\) \(y(0)=2, \quad y^{\prime}(0)=0\)