Chapter 7: The Laplace Transform

In Problems 31–34 find the given inverse Laplace transform by fnding the Laplace transform of the indicated function f.

In Problems 31–34 find the given inverse Laplace transform byfnding the Laplace transform of the indicated function f.

In Problems 31–34 find the given inverse Laplace transform by fnding the Laplace transform of the indicated function f.

In Problems 31–34 find the given inverse Laplace transform by fnding the Laplace transform of the indicated function f.

In Problems 35–44 use the Laplace transform to solve the giveninitial-value problem.

$\mathbf{35}\mathbf{.}\frac{\mathbf{dy}}{\mathbf{dt}}\mathbf{-}\mathbf{y}\mathbf{=}\mathbf{1}\mathbf{,}\mathbf{}\mathbf{}\mathbf{}\mathbf{y}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{0}$

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

$\mathbf{2}\frac{\mathbf{dy}}{\mathbf{dt}}\mathbf{+}\mathbf{y}\mathbf{=}\mathbf{0}\mathbf{,}\mathbf{y}\left(0\right)\mathbf{=}\mathbf{-}\mathbf{3}$

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

${\mathbf{y}}^{\mathbf{\text{'}}}\mathbf{+}\mathbf{6}\mathbf{y}\mathbf{=}{\mathbf{e}}^{\mathbf{4}\mathbf{t}}\mathbf{,}\mathbf{y}\left(0\right)\mathbf{=}\mathbf{2}$

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

${\mathbf{y}}^{\mathbf{\text{'}}}\mathbf{-}\mathbf{y}\mathbf{=}\mathbf{2}\mathbf{cos}\mathbf{5}\mathbf{t}\mathbf{,}\mathbf{y}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{0}$

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

${\mathbf{y}}^{\mathbf{\text{'}}\mathbf{\text{'}}}\mathbf{+}\mathbf{5}{\mathbf{y}}^{\mathbf{\text{'}}}\mathbf{+}\mathbf{4}\mathbf{y}\mathbf{=}\mathbf{0}\mathbf{,}\mathbf{y}\left(0\right)\mathbf{=}\mathbf{1}\mathbf{,}{\mathbf{y}}^{\mathbf{\text{'}}}\left(0\right)\mathbf{=}\mathbf{0}$

In Problems 1-30 use appropriate algebra and Theorem 7.2.1 to find the given inverse Laplace transform.

${\mathcal{L}}^{\mathbf{-}\mathbf{1}}\left\{\frac{1}{s^2}-\frac{48}{s^5}\right\}$