Chapter 7: Laplace Transforms

In Problems $1-14$, solve the given initial value problem using the method of Laplace transforms

$\mathbf{}\mathbf{10}\mathbf{.}\mathbf{}\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{-}\mathbf{4}\mathbf{y}\mathbf{=}\mathbf{4}\mathbf{t}\mathbf{-}\mathbf{8}{\mathbf{e}}^{\mathbf{-}\mathbf{2}\mathbf{t}}\mathbf{;}\mathbf{y}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{0}\mathbf{,}\mathbf{y}\mathbf{\text{'}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{5}$

In Problems $\mathbf{1}\mathbf{-}\mathbf{14}$ , solve the given initial value problem using the method of Laplace transforms.

$\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{-}\mathbf{y}\mathbf{=}\mathbf{t}\mathbf{-}\mathbf{2}\mathbf{;}\mathbf{y}\left(\mathbf{2}\right)\mathbf{=}\mathbf{3}\mathbf{,}\mathbf{y}\mathbf{\text{'}}\left(\mathbf{2}\right)\mathbf{=}\mathbf{0}$

In Problems $\mathbf{1}\mathbf{-}\mathbf{14}$ , solve the given initial value problem using the method of Laplace transforms

$\mathbf{w}\mathbf{\text{'}}\mathbf{\text{'}}-\mathbf{2}\mathbf{w}\mathbf{\text{'}}\mathbf{+}\mathbf{w}\mathbf{=}\mathbf{6}\mathbf{t}\mathbf{-}\mathbf{2}\mathbf{;}\mathbf{w}\left(\mathbf{-}\mathbf{1}\right)\mathbf{=}\mathbf{3}\mathbf{;}\mathbf{w}\mathbf{\text{'}}\left(\mathbf{-}\mathbf{1}\right)\mathbf{=}\mathbf{7}$

In Problems $\mathbf{1}\mathbf{-}\mathbf{14}$ , solve the given initial value problem using the method of Laplace transforms.

$\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{-}\mathbf{y}\mathbf{\text{'}}\mathbf{-}\mathbf{2}\mathbf{y}\mathbf{=}\mathbf{-}\mathbf{8}\mathbf{cost}\mathbf{-}\mathbf{2}\mathbf{sint}\mathbf{;}\mathbf{y}\left(\frac{\mathbf{\pi }}{\mathbf{2}}\right)\mathbf{=}\mathbf{1}\mathbf{,}\mathbf{y}\mathbf{\text{'}}\left(\frac{\mathbf{\pi }}{\mathbf{2}}\right)\mathbf{=}\mathbf{0}$

In Problems $\mathbf{1}\mathbf{-}\mathbf{14}$, solve the given initial value problem using the method of Laplace transforms.

$\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{y}\mathbf{=}\mathbf{t}\mathbf{;}\mathbf{y}\left(\mathbf{\pi }\right)\mathbf{=}\mathbf{0}\mathbf{,}\mathbf{y}\mathbf{\text{'}}\left(\mathbf{\pi }\right)\mathbf{=}\mathbf{0}$

In Problems $\mathbf{15}-\mathbf{24}$, solve for $\mathbf{Y}\left(\mathbf{s}\right)$ , the Laplace transform of the solution $\mathbf{y}\left(\mathbf{t}\right)$to the given initial value problem.

$\mathbf{}\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{-}\mathbf{3}\mathbf{y}\mathbf{\text{'}}\mathbf{+}\mathbf{2}\mathbf{y}\mathbf{=}\mathbf{cost}\mathbf{;}\mathbf{y}\left(\mathbf{0}\right)\mathbf{=}\mathbf{0}\mathbf{,}\mathbf{y}\mathbf{\text{'}}\left(\mathbf{0}\right)\mathbf{=}\mathbf{-}\mathbf{1}$

Question: In Problems $\mathbf{15}-\mathbf{24}$, solve for $\mathbf{Y}\left(\mathbf{s}\right)$, the Laplace transform of the solution $\mathbf{y}\left(\mathbf{t}\right)$ to the given initial value problem.

$\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{6}\mathbf{y}\mathbf{=}{\mathbf{t}}^{\mathbf{2}}\mathbf{-}\mathbf{1}\mathbf{;}\mathbf{y}\left(\mathbf{0}\right)\mathbf{=}\mathbf{0}\mathbf{,}\mathbf{y}\mathbf{\text{'}}\left(\mathbf{0}\right)\mathbf{=}\mathbf{-}\mathbf{1}$

In Problems $\mathbf{15}-\mathbf{24}$, solve for $Y\left(s\right)$, the Laplace transform of the solution $y\left(t\right)$ to the given initial value problem.

$17.y^{\text{'}\text{'}}+y^{\text{'}}-y=t^3;y\left(0\right)=1,y^{\text{'}}\left(0\right)=0$

In Problems $\mathbf{15}-\mathbf{24}$ , solve for $\mathbf{Y}\left(\mathbf{s}\right)$ , the Laplace transform of the solution $y\left(t\right)$ to the given initial value problem.

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

In Problems $\mathbf{\text{15}}\text{-}\mathbf{\text{24}}$ , solve for , the Laplace transform of the solution $\mathbf{\text{y}}\left(\mathbf{t}\right)$to the given initial value problem.