Chapter 7: The Laplace Transform

Use the Laplace transform to solve the given initial-value problem

${\mathit{y}}^{\mathbf{\text{'}}\mathbf{\text{'}}}\mathbf{+}\mathbf{9}\mathit{y}\mathbf{=}{\mathit{e}}^{\mathbf{t}}\mathbf{,}\mathit{y}\left(0\right)\mathbf{=}\mathbf{0}\mathbf{,}{\mathit{y}}^{\mathbf{\text{'}}}\left(0\right)\mathbf{=}\mathbf{0}$

In Problems 35–42 use the Laplace transform to solve the given

equation.

$\mathbf{42}\mathbf{.}{\mathbf{\int }}_{\mathbf{0}}^{\mathbf{t}}\mathbf{f}\mathbf{\left(}\mathbf{\tau }\mathbf{\right)}\mathbf{f}\mathbf{(}\mathbf{t}\mathbf{-}\mathbf{\tau }\mathbf{)}\mathbf{d}\mathbf{\tau }\mathbf{=}\mathbf{6}{\mathbf{t}}^{\mathbf{3}}$

In Problems 43-46 use Problems 41 and 42 and the fact that $\Gamma \left(\frac{1}{2}\right)=\sqrt{\pi }$to find the Laplace transform of the given function.

43.$f\left(t\right)=t^{-1/2}$

$L^{-1}\left\{\frac{e^{-2s}}{s^3}\right\}$

Use the Laplace transform to solve the given initial-value problem

$\mathbf{2}{\mathit{y}}^{\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{\text{'}}}\mathbf{+}\mathbf{3}{\mathit{y}}^{\mathbf{\text{'}}\mathbf{\text{'}}}\mathbf{-}\mathbf{3}{\mathit{y}}^{\mathbf{\text{'}}}\mathbf{-}\mathbf{2}\mathit{y}\mathbf{=}{\mathit{e}}^{\mathbf{-}\mathbf{t}}\mathbf{,}\mathit{y}\left(0\right)\mathbf{=}\mathbf{0}\mathbf{,}{\mathit{y}}^{\mathbf{\text{'}}}\left(0\right)\mathbf{=}\mathbf{0}\mathbf{,}{\mathit{y}}^{\mathbf{\text{'}}\mathbf{\text{'}}}\left(0\right)\mathbf{=}\mathbf{1}$

use the Laplace transform to solve each system

$$\begin{gathered} {\mathbf{x}}^{\mathbf{\text{'}}}\mathbf{+}\mathbf{y}\mathbf{=}\mathbf{t} \\ \mathbf{4}\mathbf{x}\mathbf{+}{\mathbf{y}}^{\mathbf{\text{'}}}\mathbf{=}\mathbf{0} \\ \mathbf{x}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{1}\mathbf{,}\mathbf{}\mathbf{}\mathbf{}\mathbf{y}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{2} \end{gathered}$$

In Problems 43-46 use Problems 41 and 42 and the fact that $\Gamma \left(\frac{1}{2}\right)=\sqrt{\pi }$to find the Laplace transform of the given function.

44.$f\left(t\right)=t^{1/2}$

Use the Laplace transform to solve the given initial-value problem

${\mathit{y}}^{\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{\text{'}}}\mathbf{+}\mathbf{2}{\mathit{y}}^{\mathbf{\text{'}}\mathbf{\text{'}}}\mathbf{-}{\mathit{y}}^{\mathbf{\text{'}}}\mathbf{-}\mathbf{2}\mathit{y}\mathbf{=}\mathit{s}\mathit{i}\mathit{n}\mathbf{3}\mathit{t}\mathbf{,}\mathit{y}\left(0\right)\mathbf{=}\mathbf{0}\mathbf{,}{\mathit{y}}^{\mathbf{\text{'}}}\left(0\right)\mathbf{=}\mathbf{0}\mathbf{,}{\mathit{y}}^{\mathbf{\text{'}}\mathbf{\text{'}}}\left(0\right)\mathbf{=}\mathbf{1}$

In Problem $E\left(t\right)=E_0{e}^{-kt},k>0.$$\mathcal{L}\left\{\right(t-1\left)\mathcal{U}\right(t-1\left)\right\}$ems $37-38$find either $F\left(s\right)$or $f\left(t\right)$, as indicated$L^{-1}\left\{\frac{{(1+e^{-2s})}^2}{s+2}\right\}$

use the Laplace transform to solve each system

$\mathbf{44}\mathbf{.}{\mathbf{x}}^{\mathbf{\text{'}}\mathbf{\text{'}}}\mathbf{+}{\mathbf{y}}^{\mathbf{\text{'}}\mathbf{\text{'}}}\mathbf{=}{\mathbf{e}}^{\mathbf{2}\mathbf{t}}\mathbf{2}{\mathbf{x}}^{\mathbf{\text{'}}}\mathbf{+}{\mathbf{y}}^{\mathbf{\text{'}}\mathbf{\text{'}}}\mathbf{=}\mathbf{-}{\mathbf{e}}^{\mathbf{2}\mathbf{t}}\mathbf{x}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{0}\mathbf{,}\mathbf{}\mathbf{}\mathbf{}\mathbf{y}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{0}\mathbf{,}{\mathbf{x}}^{\mathbf{\text{'}}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{0}\mathbf{,}\mathbf{}\mathbf{}\mathbf{}{\mathbf{y}}^{\mathbf{\text{'}}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{0}$