Chapter 7: The Laplace Transform

Explain why the function

$\mathbf{f}\left(t\right)\mathbf{=}\{\begin{array}{ll}t,& 0\le t<2\\ 4,& 2<t<5\\ \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\left(t-5\right)$}\right.,& t>5\end{array}$

is not piecewise continuous on$\mathbf{[}\mathbf{0}\mathbf{,}\mathbf{\infty }\mathbf{)}\mathbf{.}$

Show that the function$\mathbf{f}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}\mathbf{1}\mathbf{/}{\mathbf{t}}^{\mathbf{2}}$ does not possess a Laplace transform. [Hint: Write$\mathcal{L}\left\{1/t^2\right\}$ as two improper integrals:$\mathcal{L}\left\{1/t^2\right\}\mathbf{=}{\mathbf{\int }}_{\mathbf{0}}^{\mathbf{1}}\frac{{\mathbf{e}}^{\mathbf{-}\mathbf{st}}}{{\mathbf{t}}^{\mathbf{2}}}\mathbf{dt}\mathbf{+}{\mathbf{\int }}_{\mathbf{1}}^{\mathbf{\infty }}\frac{{\mathbf{e}}^{\mathbf{-}\mathbf{st}}}{{\mathbf{t}}^{\mathbf{2}}}\mathbf{dt}\mathbf{=}{\mathbf{I}}_{\mathbf{1}}\mathbf{+}{\mathbf{I}}_{\mathbf{2}}\mathbf{.}$

Show that${\mathbf{I}}_{\mathbf{1}}$ diverges.]

The function $\mathbf{f}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}\mathbf{2}{\mathbf{te}}^{{\mathbf{t}}^{\mathbf{2}}}{\mathbf{cost}}^{\mathbf{2}}$ is not of exponential order. Nevertheless, show that the Laplace transform$\mathcal{L}\left\{2{\mathrm{te}}^{t^2}{\mathrm{cose}}^{t^2}\right\}$ exists. [Hint: Start with integration by parts.]

If $\mathcal{L}\mathbf{\left\{}\mathbf{f}\mathbf{\right(}\mathbf{t}\mathbf{\left)}\mathbf{\right\}}\mathbf{=}\mathbf{F}\mathbf{\left(}\mathbf{s}\mathbf{\right)}$ and$\mathbf{a}\mathbf{>}\mathbf{0}$ is a constant, show that

$\mathcal{L}\mathbf{\left\{}\mathbf{f}\mathbf{\right(}\mathbf{at}\mathbf{\left)}\mathbf{\right\}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{a}}\mathbf{F}\left(\frac{s}{a}\right)$

This result is known as the change of scale theorem.

In Problems 55-58 use the given Laplace transform and the result in Problem 54 to find the indicated Laplace transform. Assume that a and k are positive constants.

55.$\mathcal{L}\left\{e^t\right\}\mathbf{=}\frac{\mathbf{1}}{\mathbf{s}\mathbf{-}\mathbf{1}}\mathbf{;}\mathbf{}\mathbf{}\mathbf{}\mathcal{L}\left\{e^{\mathrm{at}}\right\}\mathbf{}\mathbf{}\mathbf{}$

In Problems 55-58 use the given Laplace transform and the result in Problem 54 to find the indicated Laplace transform. Assume that a and k are positive constants.

56.$\mathcal{L}\mathbf{\left\{}\mathbf{sint}\mathbf{\right\}}\mathbf{=}\frac{\mathbf{1}}{{\mathbf{s}}^{\mathbf{2}}\mathbf{+}\mathbf{1}}\mathbf{;}\mathbf{}\mathbf{}\mathbf{}\mathcal{L}\mathbf{\left\{}\mathbf{sinkt}\mathbf{\right\}}$

In Problems 55-58 use the given Laplace transform and the result in Problem 54 to find the indicated Laplace transform. Assume that a and k are positive constants.

57.$\mathcal{L}\mathbf{\{}\mathbf{1}\mathbf{-}\mathbf{cost}\mathbf{\}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{s}\left(s^2+1\right)}\mathbf{;}\mathbf{}\mathbf{}\mathbf{}\mathcal{L}\mathbf{\{}\mathbf{1}\mathbf{-}\mathbf{coskt}\mathbf{\}}$

In Problems 55-58 use the given Laplace transform and the result in Problem 54 to find the indicated Laplace transform. Assume that a and k are positive constants.

58.$\mathcal{L}\mathbf{\left\{}\mathbf{sintsinht}\mathbf{\right\}}\mathbf{=}\frac{\mathbf{2}\mathbf{s}}{{\mathbf{s}}^{\mathbf{4}}\mathbf{+}\mathbf{4}}\mathbf{;}\mathbf{}\mathcal{L}\mathbf{\left\{}\mathbf{sinktsinhkt}\mathbf{\right\}}$

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

15. ${\mathit{L}}^{\mathbf{-}\mathbf{1}}\left\{\frac{1}{{\left(s-5\right)}^3}\right\}$=……

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

16. ${\mathit{L}}^{\mathbf{-}\mathbf{1}}\left\{\frac{1}{\left(s^2-5\right)}\right\}\mathbf{=}$……