91Ó°ÊÓ

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.

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

In Problems 11–20, determine the partial fraction expansion for the given rational function.$\frac{{\mathbf{s}}^{\mathbf{2}}\mathbf{−}\mathbf{26}\mathbf{s}\mathbf{−}\mathbf{47}}{\mathbf{(}\mathbf{s}\mathbf{−}\mathbf{1}\mathbf{)}\mathbf{(}\mathbf{s}\mathbf{+}\mathbf{2}\mathbf{)}\mathbf{(}\mathbf{s}\mathbf{+}\mathbf{5}\mathbf{)}}$

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.

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

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.

$\begin{array}{l}{\mathbf{\text{x}}}^{\mathbf{\text{'}}}\mathbf{\text{+²â=³æ; â¶Ä‰â¶Ä‰x(0)=0,²â(0)=1}}\\ {\mathbf{\text{2x}}}^{\mathbf{\text{'}}}{\mathbf{\text{+y}}}^{\mathbf{\text{''}}}{\mathbf{\text{=u(t-3);y}}}^{\mathbf{\text{'}}}\mathbf{\text{(0)=-1}}\end{array}$

In Problems 11–20, determine the partial fraction expansion for the given rational function.

$\frac{\mathbf{−}\mathbf{s}\mathbf{−}\mathbf{7}}{\mathbf{(}\mathbf{s}\mathbf{+}\mathbf{1}\mathbf{)}\mathbf{(}\mathbf{s}\mathbf{−}\mathbf{2}\mathbf{)}}$

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 11–20, determine the partial fraction expansion for the given rational function.$\frac{\mathbf{−}\mathbf{2}{\mathbf{s}}^{\mathbf{2}}\mathbf{−}\mathbf{3}\mathbf{s}\mathbf{−}\mathbf{2}}{\mathbf{s}{\mathbf{(}\mathbf{s}\mathbf{+}\mathbf{1}\mathbf{)}}^{\mathbf{2}}}$

Use the Laplace transform table and the linearity of the Laplace transform to determine the following transforms.

\(L\left\{ {6{e^{ - 3t}} - {t^2} + 2t - 8} \right\}\)

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.

$\begin{array}{c}{\mathbf{\text{x}}}^{\mathbf{\text{'}}}{\mathbf{\text{-y}}}^{\mathbf{\text{'}}}\mathbf{\text{=(²õ¾±²Ô³Ù)³Ü(³Ù-Ï€); â¶Ä‰â¶Ä‰x(0)=1,}}\\ {\mathbf{\text{x+y}}}^{\mathbf{\text{'}}}\mathbf{\text{=0; â¶Ä‰â¶Ä‰y(0)=1}}\end{array}$

Find the Laplace transform of$\mathit{f}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}\underset{0}{\overset{t}{∫}}(t−v)e^{3v}dv$