Laplace Transforms

In Problems 3–10, determine the Laplace transform of the given function.

${\mathit{e}}^{\mathbf{2}\mathbf{t}}\mathbf{-}{\mathit{t}}^{\mathbf{3}}\mathbf{+}{\mathit{t}}^{\mathbf{2}}\mathbf{-}\mathit{s}\mathit{i}\mathit{n}\mathbf{5}\mathit{t}$

In Problems 3–10, determine the Laplace transform of the given function.

$\mathbf{7}{\mathit{e}}^{\mathbf{2}\mathbf{t}}\mathit{c}\mathit{o}\mathit{s}\mathbf{3}\mathit{t}\mathbf{-}\mathbf{2}{\mathit{e}}^{\mathbf{7}\mathbf{t}}\mathit{s}\mathit{i}\mathit{n}\mathbf{5}\mathit{t}$

In Problems 3–10, determine the Laplace transform of the given function.

$7e^{2t}cos3t-2e^{7t}sin5t$

In Problems 3-10, determine the Laplace transform of the given function.

${\mathbf{(}\mathbf{t}\mathbf{+}\mathbf{3}\mathbf{)}}^{\mathbf{2}}\mathbf{-}{\mathbf{(}{\mathbf{e}}^{\mathbf{t}}\mathbf{+}\mathbf{3}\mathbf{)}}^{\mathbf{2}}$

In Problems 3-10, determine the Laplace transform of the given function.${\mathit{t}}^{\mathbf{2}}\mathit{u}\mathbf{(}\mathit{t}\mathbf{-}\mathbf{4}\mathbf{)}$

In Problems 3-10, determine the Laplace transform of the given function.$\mathit{f}\mathbf{\left(}\mathit{t}\mathbf{\right)}\mathbf{=}\mathit{c}\mathit{o}\mathit{s}\mathit{t}\mathbf{,}\mathbf{-}\mathit{\pi }\mathbf{/}\mathbf{2}\mathbf{\le }\mathit{t}\mathbf{\le }\mathit{\pi }\mathbf{/}\mathbf{2}$and $\mathit{f}\mathbf{\left(}\mathit{t}\mathbf{\right)}$has period $\mathit{\pi }$

In Problems 1–19, use the method of Laplace transforms to solve the given initial value problem. Here x′, y′, etc., denotes differentiation with respect to t; so does the symbol D.

11.

$\begin{array}{ll}{\mathit{x}}^{\mathit{\text{'}}}\mathit{+}\mathit{y}\mathit{=}\mathit{1}\mathit{-}\mathit{u}\mathit{(}\mathit{t}\mathit{-}\mathit{2}\mathit{)}\mathit{;}& \mathit{x}\mathit{\left(}\mathit{0}\mathit{\right)}\mathit{=}\mathit{0}\mathit{,}\\ \mathit{x}\mathit{+}{\mathit{y}}^{\mathit{\text{'}}}\mathit{=}\mathit{0}\mathit{;}& \mathit{y}\mathit{\left(}\mathit{0}\mathit{\right)}\mathit{=}\mathit{0}\end{array}$

Determine the inverse Laplace transform of the given function.

$\frac{\mathbf{2}\mathbf{s}\mathbf{-}\mathbf{1}}{{\mathbf{s}}^{\mathbf{2}}\mathbf{-}\mathbf{4}\mathbf{s}\mathbf{+}\mathbf{6}}$

In Problems 3-10, determine the Laplace transform of the given function.$\frac{\mathbf{4}{\mathbf{s}}^{\mathbf{2}}\mathbf{+}\mathbf{13}\mathbf{s}\mathbf{+}\mathbf{19}}{\mathbf{(}\mathbf{s}\mathbf{-}\mathbf{1}\mathbf{)}\mathbf{(}{\mathbf{s}}^{\mathbf{2}}\mathbf{+}\mathbf{4}\mathbf{s}\mathbf{+}\mathbf{13}\mathbf{)}}$

Determine the inverse Laplace transform of the given function.

$\frac{{\mathbf{s}}^{\mathbf{2}}\mathbf{+}\mathbf{16}\mathbf{s}\mathbf{+}\mathbf{9}}{\mathbf{(}\mathbf{s}\mathbf{+}\mathbf{1}\mathbf{)}\mathbf{(}\mathbf{s}\mathbf{+}\mathbf{3}\mathbf{)}\mathbf{(}\mathbf{s}\mathbf{-}\mathbf{2}\mathbf{)}}$

Determine the inverse Laplace transform of the given function.

 $\frac{\mathbf{2}{\mathbf{s}}^{\mathbf{2}}\mathbf{+}\mathbf{3}\mathbf{s}\mathbf{-}\mathbf{1}}{{\mathbf{(}\mathbf{s}\mathbf{+}\mathbf{1}\mathbf{)}}^{\mathbf{2}}\mathbf{(}\mathbf{s}\mathbf{+}\mathbf{2}\mathbf{)}}$

In Problems 3-10, determine the Laplace transform of the given function.$\frac{\mathbf{1}}{{\mathbf{(}{\mathbf{s}}^{\mathbf{2}}\mathbf{+}\mathbf{9}\mathbf{)}}^{\mathbf{2}}}$