Chapter 7: The Laplace Transform

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

17. ${\mathit{L}}^{\mathbf{-}\mathbf{1}}\left\{\frac{s}{s^2-10s+29}\right\}$=……

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

18.${\mathit{L}}^{\mathbf{-}\mathbf{1}}\left\{\frac{e^{-5s}}{s^2}\right\}\mathbf{=}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}$

${\mathit{L}}^{\mathbf{-}\mathbf{1}}\left\{\frac{s+\pi }{s^2+{\pi }^2}{e}^{-s}\right\}\mathbf{=}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}$In Problems 3–24 fill in the blanks or answer true or false.

19.

Transform of the Logarithm Because f(t) 5 lnt has an innitediscontinuity at t 5 0 it might be assumed that $\mathbf{L}\mathbf{\left\{}\mathbf{lnt}\mathbf{\right\}}$does not exist; however, this is incorrect. The point of this problem is to guide you through the formal steps leading to the Laplace transform of$\mathbf{f}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}\mathbf{lnt}\mathbf{,}\mathbf{t}\mathbf{>}\mathbf{0}$

a) Use integration by parts to show that$\mathbf{L}\mathbf{\left[}\mathbf{lnt}\mathbf{\right]}\mathbf{=}\mathbf{sL}\mathbf{\left[}\mathbf{tlnt}\mathbf{\right]}\frac{\mathbf{1}}{\mathbf{s}\mathbf{*}}$

b) If$\mathbf{L}\mathbf{\left[}\mathbf{lnt}\mathbf{\right]}\mathbf{=}\mathbf{Y}\mathbf{\left(}\mathbf{s}\mathbf{\right)}$, use Theorem 7.4.1 with n = 1 to show that part (a) becomes$\mathbf{s}\frac{\mathbf{dY}}{\mathbf{ds}}\mathbf{+}\mathbf{Y}\mathbf{=}\mathbf{-}\frac{\mathbf{1}}{\mathbf{s}}$Find an explicit solution Y(s) of the foregoing differential equation.

c) Finally, the integral denition of Euler’s constant (sometimes called the Euler-Mascheroni constant) is$\mathit{\gamma }\mathbf{=}\mathbf{-}\mathbf{\int }{\mathit{t}}_{\mathbf{0}}^{\mathbf{x}}{\mathit{e}}^{\mathbf{-}\mathbf{t}}\mathit{l}\mathit{n}\mathit{t}\mathit{d}\mathit{t}\mathbf{,}\mathbf{}\mathbf{}\mathit{w}\mathit{h}\mathit{e}\mathit{r}\mathit{e}\mathbf{}\mathit{\gamma }\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{5772156649}\mathbf{\dots }$

Use$\mathbf{Y}\mathbf{\left(}\mathbf{1}\mathbf{\right)}\mathbf{=}\mathbf{-}\mathbf{\gamma }$in the solution in part (b) to show that $\mathbf{L}\mathbf{\left\{}\mathbf{lnt}\mathbf{\right\}}\mathbf{=}\mathbf{-}\frac{\mathbf{\gamma }}{\mathbf{s}}\mathbf{-}\frac{\mathbf{lns}}{\mathbf{s}}\mathbf{,}\mathbf{}\mathbf{s}\mathbf{>}\mathbf{0}$

Suppose a 32pound weight stretches a spring 2feet. If the weight is released from rest at the equilibrium position, find the equation of motion$x\left(t\right)$if an impressed force$f\left(t\right)=20t$acts on the system for$0⩽t<5$and is then removed (see Example 5). Ignore any damping forces. Use a graphing utility to graphdata-custom-editor="chemistry" $x\left(t\right)$on the interval$[0,10]$.

In Problems 1 and 2 use, the definition of the Laplace transform to find

$\mathit{f}\mathbf{\left(}\mathit{t}\mathbf{\right)}\mathbf{}\mathbf{=}\mathbf{}\{\begin{array}{ll}t& 0\le t<1\\ 2t& t\ge 1\end{array}$

In Problems 3 - 24 fill in the blanks or answer true or false. $\mathbf{20}\mathbf{.}{\mathit{L}}^{\mathbf{-}\mathbf{1}}\left\{\frac{1}{L^2{s}^2+n^2{\pi }^2}\right\}\mathbf{=}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}$

10. In Problems 1-30 use appropriate algebra and Theorem 7.2.1 to find the given inverse Laplace transform.

${\mathcal{L}}^{\mathbf{-}\mathbf{1}}\left\{\frac{1}{5s-2}\right\}$

11. In Problems 1-30 use appropriate algebra and Theorem 7.2.1 to find the given inverse Laplace transform.

${\mathcal{L}}^{\mathbf{-}\mathbf{1}}\left\{\frac{5}{s^2+49}\right\}$

12. In Problems 1-30 use appropriate algebra and Theorem 7.2.1 to find the given inverse Laplace transform.

${\mathcal{L}}^{\mathbf{-}\mathbf{1}}\left\{\frac{10s}{s^2+16}\right\}$