Chapter 7: The Laplace Transform

In problems 1-20 find either f(S) or f(t), as indicated $\mathit{L}\left\{e^{2t}{(t-1)}^2\right\}$

If we assume that $\mathbf{L}\mathbf{\left[}\mathbf{f}\mathbf{\right(}\mathbf{t}\mathbf{)}\mathbf{/}\mathbf{t}\mathbf{]}$exists and $\mathbf{L}\mathbf{\left[}\mathbf{f}\mathbf{\right(}\mathbf{t}\mathbf{\left)}\mathbf{\right]}\mathbf{=}\mathbf{F}\mathbf{\left(}\mathbf{s}\mathbf{\right)}$then $\mathit{L}\left\{\frac{f\left(t\right)}{t}\right\}\mathbf{=}{\mathbf{\int }}_{\mathbf{s}}^{\mathbf{x}}\mathit{F}\mathbf{\left(}\mathit{u}\mathbf{\right)}\mathit{d}\mathit{u}\mathbf{.}$Use this result to and the Laplace transform of the given function. The symbols a and k are positive constants.

$\mathbf{}\mathbf{\left(}\mathbf{a}\mathbf{\right)}\mathbf{}\mathbf{f}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}\frac{\mathbf{sinat}}{\mathbf{t}}$

$\left(b\right)\mathbf{}\mathbf{f}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}\frac{\mathbf{2}\mathbf{(}\mathbf{1}\mathbf{-}\mathbf{coskt}\mathbf{)}}{\mathbf{t}}$

In problems, use the Laplace transform to solve the given initial-value problem.

$$\begin{gathered} y\text{'}\text{'}+4y\text{'}+3y=1-u(t-2)-u(t-4)+u(t-6) \\ y\left(0\right)=0,y\text{'}\left(0\right)=0 \end{gathered}$$

In Problems 3 - 24 fill in the blanks or answer true or false.

10.$\mathit{L}\left\{e^{-3t}sin 2t\right\}$
=……..

In Problems 3 - 24 fill in the blanks or answer true or false.

11.$\mathit{L}\left\{tsin 2t\right\}$ =………

In Problems 3 - 24 fill in the blanks or answer true or false.

12.$\mathit{L}\left\{sin\left(2t\right)U\left(t-\pi \right)\right\}$=…….

In Problems 3 - 24 fill in the blanks or answer true or false.

13. $\mathit{L}{}^{\mathbf{-}\mathbf{1}}\left\{\frac{20}{s^6}\right\}$=….

Use part (c) of Theorem 7.1.1 to show that

$\mathcal{L}\left\{e^{(a+\mathrm{ib})t}\right\}\mathbf{=}\frac{\mathbf{s}\mathbf{-}\mathbf{a}\mathbf{+}\mathbf{ib}}{{\mathbf{(}\mathbf{s}\mathbf{-}\mathbf{a}\mathbf{)}}^{\mathbf{2}}\mathbf{+}{\mathbf{b}}^{\mathbf{2}}}\mathbf{,}$

where a and b are real and ${\mathbf{i}}^{\mathbf{2}}\mathbf{=}\mathbf{-}\mathbf{1}\mathbf{.}$Show how Euler's formula (page 136) can then be used to deduce the results

$$\begin{gathered} \mathcal{L}\left\{e^{\mathrm{at}}\mathrm{cosbt}\right\}\mathbf{=}\frac{\mathbf{s}\mathbf{-}\mathbf{a}}{{\mathbf{(}\mathbf{s}\mathbf{-}\mathbf{a}\mathbf{)}}^{\mathbf{2}}\mathbf{+}{\mathbf{b}}^{\mathbf{2}}} \\ \mathcal{L}\left\{e^{\mathrm{at}}\mathrm{sinbt}\right\}\mathbf{=}\frac{\mathbf{b}}{{\mathbf{(}\mathbf{s}\mathbf{-}\mathbf{a}\mathbf{)}}^{\mathbf{2}}\mathbf{+}{\mathbf{b}}^{\mathbf{2}}}\mathbf{.} \end{gathered}$$

In Problems 3 - 24 fill in the blanks or answer true or false.

14.${\mathit{L}}^{\mathbf{-}\mathbf{1}}\left\{\frac{1}{3s-1}\right\}$ =…….

Under what conditions is a linear function$\mathbf{f}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{=}\mathbf{mx}\mathbf{+}\mathbf{b}\mathbf{,}\mathbf{m}\mathbf{\ne }\mathbf{0}\mathbf{,}$ a linear transform?