Laplace Transforms

In Problems 1-20, determine the Laplace transform of the given function using Table 7.1 on page 356 and the properties of the transform given in Table 7.2. [Hint: In Problems 12-20, use an appropriate trigonometric identity.]

$\mathbf{(}\mathbf{t}\mathbf{-}\mathbf{1}{\mathbf{)}}^{\mathbf{4}}$.

In Problems 1-20, determine the Laplace transform of the given function using Table 7.1 on page 356 and the properties of the transform given in Table 7.2. [Hint: In Problems 12-20, use an appropriate trigonometric identity.]

${\left(1+e^{-t}\right)}^{\mathbf{2}}$.

In Problems 1-20, determine the Laplace transform of the given function using Table 7.1 on page 356 and the properties of the transform given in Table 7.2. [Hint: In Problems 12-20, use an appropriate trigonometric identity.]

${\mathbf{e}}^{\mathbf{-}\mathbf{t}}\mathbf{tsin}\mathbf{2}\mathbf{t}$

In Problems 1-20, determine the Laplace transform of the given function using Table 7.1 on page 356 and the properties of the transform given in Table 7.2. [Hint: In Problems 12-20, use an appropriate trigonometric identity.]

${\mathbf{te}}^{\mathbf{2}\mathbf{t}}\mathbf{cos}\mathbf{5}\mathbf{t}$

In Problems 1-20, determine the Laplace transform of the given function using Table 7.1 on page 356 and the properties of the transform given in Table 7.2. [Hint: In Problems 12-20, use an appropriate trigonometric identity.]

$\mathbf{coshbt}$

In Problems 1-20, determine the Laplace transform of the given function using Table 7.1 on page 356 and the properties of the transform given in Table 7.2. [Hint: In Problems 12-20, use an appropriate trigonometric identity.]

$\mathbf{\text{sin3t cos3t}}$

In Problems 1-20, determine the Laplace transform of the given function using Table 7.1 on page 356 and the properties of the transform given in Table 7.2. [Hint: In Problems 12-20, use an appropriate trigonometric identity.]

${\mathbf{sin}}^{\mathbf{2}}\mathbf{t}$.

In Problems 1-20, determine the Laplace transform of the given function using Table 7.1 on page 356 and the properties of the transform given in Table 7.2. [Hint: In Problems 12-20, use an appropriate trigonometric identity.]

${\mathbf{e}}^{\mathbf{7}\mathbf{t}}{\mathbf{sin}}^{\mathbf{2}}\mathbf{t}$

In Problems 1-20, determine the Laplace transform of the given function using Table 7.1 on page 356 and the properties of the transform given in Table 7.2. [Hint: In Problems 12-20, use an appropriate trigonometric identity.]

${\mathbf{cos}}^{\mathbf{3}}\mathbf{t}$

In Problems 1-20, determine the Laplace transform of the given function using Table 7.1 on page 356 and the properties of the transform given in Table 7.2. [Hint: In Problems 12-20, use an appropriate trigonometric identity.]

${\mathbf{tsin}}^{\mathbf{2}}\mathbf{t}$

In Problems 1-20, determine the Laplace transform of the given function using Table 7.1 on page 356 and the properties of the transform given in Table 7.2. [Hint: In Problems 12-20, use an appropriate trigonometric identity.]

$\mathbf{sin}\mathbf{2}\mathbf{t}\mathbf{ }\mathbf{sin}\mathbf{5}\mathbf{t}$

In Problems 1-20, determine the Laplace transform of the given function using Table 7.1 on page 356 and the properties of the transform given in Table 7.2. [Hint: In Problems 12-20, use an appropriate trigonometric identity.]

$\mathbf{cos}\mathbf{ }\mathbf{nt}\mathbf{ }\mathbf{cos}\mathbf{ }\mathbf{mt}\mathbf{ }\mathbf{,}\mathbf{ }\mathbf{m}\mathbf{ }\mathbf{ }\mathbf{\ne }\mathbf{n}$

Given that $\mathcal{L}\mathbf{\left\{}\mathbf{cosbt}\mathbf{\right\}}\mathbf{\left(}\mathbf{s}\mathbf{\right)}\mathbf{=}\mathbf{s}\mathbf{/}\left(s^2+b^2\right)$, use the translation property to compute $\mathcal{L}\left\{e^{\mathrm{at}}\mathrm{cosbt}\right\}$.

Use Theorem 4 on page362 to show how entry 32 follows from entry 31 in the Laplace transform table on the inside back cover of the text.

The transfer function of a linear system is defined as the ratio of the Laplace transform of the output function y(t) to the Laplace transform of the input function g(t), when all initial conditions are zero. If a linear system is governed by the differential equation

$\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{+}\mathbf{6}\mathbf{y}\mathbf{\text{'}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{+}\mathbf{10}\mathbf{y}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}\mathbf{g}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{,}\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{t}\mathbf{>}\mathbf{0}$

use the linearity property of the Laplace transform and Theorem 5 on page363 on the Laplace transform of higher-order derivatives to determine the transfer function $\mathbf{H}\mathbf{\left(}\mathbf{s}\mathbf{\right)}\mathbf{=}\mathbf{Y}\mathbf{\left(}\mathbf{s}\mathbf{\right)}\mathbf{/}\mathbf{G}\mathbf{\left(}\mathbf{s}\mathbf{\right)}$of this system.

Determine the inverse Laplace transform of the given function.

$\frac{\mathbf{1}}{{\mathbf{s}}^{\mathbf{2}}\mathbf{+}\mathbf{4}\mathbf{s}\mathbf{+}\mathbf{8}}$.

In Problems 1-20, determine the Laplace transform of the given function using Table 7.1 on page 356 and the properties of the transform given in Table 7.2. [Hint: In Problems 12-20, use an appropriate trigonometric identity.]

$\mathbf{cosnt}\mathbf{ }\mathbf{sinmt}\mathbf{,}\mathbf{m}\mathbf{\ne }\mathbf{n}$

In Problems 1-20, determine the Laplace transform of the given function using Table 7.1 on page 356 and the properties of the transform given in Table 7.2. [Hint: In Problems 12-20, use an appropriate trigonometric identity.]

$\mathbf{t}\mathbf{ }\mathbf{sin}\mathbf{2}\mathbf{t}\mathbf{ }\mathbf{sin}\mathbf{ }\mathbf{5}\mathbf{t}$

Find the transfer function, as defined in Problem 29, for the linear system governed by

$\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{+}\mathbf{5}\mathbf{y}\mathbf{\text{'}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{+}\mathbf{6}\mathbf{y}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}\mathbf{g}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{,}\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{t}\mathbf{>}\mathbf{0}$.

In Problems 1-10, determine the inverse Laplace transform of the given function.

 $\frac{\mathbf{3}\mathbf{s}\mathbf{-}\mathbf{15}}{\mathbf{2}{\mathbf{s}}^{\mathbf{2}}\mathbf{-}\mathbf{4}\mathbf{s}\mathbf{+}\mathbf{10}}$

Determine the inverse Laplace transform of the given function.

$\frac{\mathbf{s}\mathbf{+}\mathbf{1}}{{\mathbf{s}}^{\mathbf{2}}\mathbf{+}\mathbf{2}\mathbf{s}\mathbf{+}\mathbf{10}}$.

Determine the inverse Laplace transform of the given function.

$\frac{\mathbf{2}\mathbf{s}\mathbf{+}\mathbf{16}}{{\mathbf{s}}^{\mathbf{2}}\mathbf{+}\mathbf{4}\mathbf{s}\mathbf{+}\mathbf{13}}$.

Determine the inverse Laplace transform of the given function.

$\frac{\mathbf{1}}{{\mathbf{s}}^{\mathbf{5}}}$.

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

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

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

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

$\frac{\mathbf{-}\mathbf{8}{\mathbf{s}}^{\mathbf{2}}\mathbf{-}\mathbf{5}\mathbf{s}\mathbf{+}\mathbf{9}}{\mathbf{(}\mathbf{s}\mathbf{+}\mathbf{1}\mathbf{)}\mathbf{(}{\mathbf{s}}^{\mathbf{2}}\mathbf{-}\mathbf{3}\mathbf{s}\mathbf{+}\mathbf{2}\mathbf{)}}$

Determine the inverse Laplace transform of the given function.

\(\frac{2}{{{s^2} + 4}}\)

Determine the inverse Laplace transform of the given function.

\(\frac{4}{{{s^2} + 9}}\)

Determine the inverse Laplace transform of the given function.

\(\frac{3}{{{{\left( {2s + 5} \right)}^3}}}\).

Determine the inverse Laplace transform of the given function.

$\frac{\mathbf{6}}{{\mathbf{(}\mathbf{s}\mathbf{-}\mathbf{1}\mathbf{)}}^{\mathbf{4}}}$.

In Problems 1-10, determine the inverse Laplace transform of the given function.

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

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 11–20, determine the partial fraction expansions for the given rational function.

\(\frac{{ - 5s - 36}}{{\left( {s + 2} \right)\left( {{s^2} + 9} \right)}}\)

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

\(\frac{{3s + 5}}{{s\left( {{s^2} + s - 6} \right)}}\)

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

\(\frac{{3{s^2} + 5s + 3}}{{{s^4} + {s^3}}}\)

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

\(\frac{1}{{(s - 3)\left( {{s^2} + 2s + 2} \right)}}\)

In Problems \(21 - 30\), determine \({\mathcal{L}^{ - 1}}\left\{ F \right\}\).

\(F\left( s \right) = \frac{{s + 11}}{{\left( {s - 1} \right)\left( {s + 3} \right)}}\)

In Problems\(21 - 30\), determine \({\mathcal{L}^{ - 1}}\{ F\} \).

\(F\left( s \right) = \frac{{5{s^2} + 34s + 53}}{{{{\left( {s + 3} \right)}^2}\left( {s + 1} \right)}}\)

In Problems \(21 - 30\), determine \({\mathcal{L}^{ - 1}}\left\{ F \right\}\).

\(F\left( s \right) = \frac{{7{s^2} + 23s + 30}}{{\left( {s - 2} \right)\left( {{s^2} + 2s + 5} \right)}}\)

In Problems\(21 - 30\), determine \({\mathcal{L}^{ - 1}}\{ F\} \).

\(F(s) = \frac{{7{s^3} - 2{s^2} - 3s + 6}}{{{s^3}(s - 2)}}\)

solve the given initial value problem using the method of Laplace transforms. Sketch the graph of the solution.

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

solve the given initial value problem using the method of Laplace transforms. Sketch the graph of the solution.

$\mathit{w}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathit{w}\mathbf{=}\mathit{u}\mathbf{(}\mathbf{t}\mathbf{-}\mathbf{-}\mathbf{2}\mathbf{)}\mathbf{-}\mathbf{-}\mathit{u}\mathbf{(}\mathbf{t}\mathbf{-}\mathbf{-}\mathbf{4}\mathbf{)}\mathbf{;}\mathit{w}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{1}\mathbf{,}\mathit{w}\mathbf{\text{'}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{0}$ 

solve the given initial value problem using the method of Laplace transforms. Sketch the graph of the solution.$\mathit{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathit{y}\mathbf{=}\mathit{t}\mathbf{-}\mathbf{-}\left(t--4\right)\mathit{u}\left(t--2\right)\mathbf{;}\mathit{y}\left(0\right)\mathbf{=}\mathbf{0}\mathbf{,}\mathit{y}\mathbf{\text{'}}\left(0\right)\mathbf{=}\mathbf{1}$

solve the given initial value problem using the method of Laplace transforms. Sketch the graph of the solution.$\mathit{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathit{y}\mathbf{=}\mathbf{3}\mathit{s}\mathit{i}\mathit{n}\mathbf{2}\mathit{t}\mathbf{-}\mathbf{-}\mathbf{3}\left(sin2t\right)\mathit{u}\mathbf{(}\mathit{t}\mathbf{-}\mathbf{2}\mathit{\pi }\mathbf{)}\mathbf{;}\mathit{y}\left(0\right)\mathbf{=}\mathbf{1}\mathbf{,}\mathit{y}\mathbf{\text{'}}\left(0\right)\mathbf{=}\mathbf{-}\mathbf{2}$

solve the given initial value problem using the method of Laplace transforms.

$\mathit{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{2}\mathit{y}\mathbf{\text{'}}\mathbf{+}\mathbf{2}\mathit{y}\mathbf{=}\mathit{u}\mathbf{(}\mathit{t}\mathbf{-}\mathbf{2}\mathbf{\pi }\mathbf{)}\mathbf{-}\mathbf{-}\mathit{u}\mathbf{(}\mathit{t}\mathbf{-}\mathbf{4}\mathbf{\pi }\mathbf{)}\mathbf{;}\mathit{y}\left(0\right)\mathbf{=}\mathbf{1}\mathbf{,}\mathit{y}\mathbf{\text{'}}\left(0\right)\mathbf{=}\mathbf{1}$

In Problems 25 - 32, solve the given initial value problem using the method of Laplace transforms.

$$\begin{gathered} \mathit{z}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{3}\mathit{z}\mathbf{\text{'}}\mathbf{+}\mathbf{2}\mathit{z}\mathbf{=}{\mathit{e}}^{\mathbf{-}\mathbf{3}\mathbf{t}}\mathit{u}\mathbf{(}\mathit{t}\mathbf{-}\mathbf{2}\mathbf{)} \\ \mathit{z}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{2}\mathbf{,}\mathbf{ }\mathit{z}\mathbf{\text{'}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{-}\mathbf{3} \end{gathered}$$

In Problems 25 - 32, solve the given initial value problem using the method of Laplace transforms.

$$\begin{gathered} \mathit{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{5}\mathit{y}\mathbf{\text{'}}\mathbf{+}\mathbf{6}\mathit{y}\mathbf{=}\mathit{t}\mathit{u}\mathbf{(}\mathit{t}\mathbf{-}\mathbf{2}\mathbf{)} \\ \mathit{y}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{0}\mathbf{ }\mathit{y}\mathbf{\text{'}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{1} \end{gathered}$$

In Problems 25 - 32, solve the given initial value problem using the method of Laplace transforms.

$$\begin{gathered} \mathit{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{4}\mathit{y}\mathbf{=}\mathit{g}\mathbf{\left(}\mathit{t}\mathbf{\right)}\mathbf{;}\mathbf{ }\mathit{y}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{1}\mathbf{,}\mathbf{ }\mathit{y}\mathbf{\text{'}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{3}\mathbf{,} \\ \mathit{w}\mathit{h}\mathit{e}\mathit{r}\mathit{e}\mathbf{ }\mathit{g}\mathbf{\left(}\mathit{t}\mathbf{\right)}\mathbf{=}\{\begin{array}{l}sint, 0⩽t⩽2\pi ,\\ 0, 2\pi <t\end{array} \end{gathered}$$

In Problems 25 - 32, solve the given initial value problem using the method of Laplace transforms.

$$\begin{gathered} \mathit{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{2}\mathit{y}\mathbf{\text{'}}\mathbf{+}\mathbf{10}\mathit{y}\mathbf{=}\mathit{g}\mathbf{\left(}\mathit{t}\mathbf{\right)}\mathbf{;}\mathbf{ } \\ \mathit{y}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{-}\mathbf{1}\mathbf{,}\mathbf{ }\mathit{y}\mathbf{\text{'}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{0}\mathbf{,}\mathbf{ } \\ \mathit{w}\mathit{h}\mathit{e}\mathit{r}\mathit{e}\mathbf{ }\mathit{g}\mathbf{\left(}\mathit{t}\mathbf{\right)}\mathbf{=}\{\begin{array}{l}10, 0⩽t⩽10,\\ 20 ,10<t<20,\\ 0, 20<t\end{array} \end{gathered}$$

In Problems 25 - 32, solve the given initial value problem using the method of Laplace transforms.

$$\begin{gathered} \mathit{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{5}\mathit{y}\mathbf{\text{'}}\mathbf{+}\mathbf{6}\mathit{y}\mathbf{=}\mathit{g}\mathbf{\left(}\mathit{t}\mathbf{\right)}\mathbf{:} \\ \mathit{y}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{0}\mathbf{,}\mathit{y}\mathbf{\text{'}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{2}\mathbf{,} \\ \mathbf{ }\mathit{w}\mathit{h}\mathit{e}\mathit{r}\mathit{e}\mathbf{ }\mathit{g}\mathbf{\left(}\mathit{t}\mathbf{\right)}\mathbf{=}\{\begin{array}{l}0 , 0⩽t<1,\\ t , 1<t<5,\\ 1 ,5<t\end{array} \end{gathered}$$