Chapter 7: The Laplace Transform

In Problems 51 and 52 solve equation (10) subject to $\mathit{i}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{}\mathbf{=}\mathbf{}\mathbf{0}$with L, R, C, and E(t) as given. Use a graphing utility to graph the solution for$\mathbf{0}\mathbf{⩽}\mathit{t}\mathbf{⩽}\mathbf{3}$.

$$\begin{gathered} \mathit{L}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{005}\mathit{h}\mathbf{,}\mathit{R}\mathbf{=}\mathbf{1}\mathit{\Omega }\mathbf{,}\mathit{C}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{02}\mathit{f} \\ \mathit{E}\mathbf{\left(}\mathit{t}\mathbf{\right)}\mathbf{=}\mathbf{100}\mathbf{[}\mathit{t}\mathbf{-}\mathbf{(}\mathit{t}\mathbf{-}\mathbf{1}\mathbf{\left)}\mathit{U}\mathbf{\right(}\mathit{t}\mathbf{-}\mathbf{1}\mathbf{\left)}\mathbf{\right]} \end{gathered}$$

In Problems 40–54, match the given graph with one of the functions in (a)-(f). The graph is given as Figure 7.3.11.

$$\begin{gathered} a)f(t)-f(t\left)u\right(t-a\left) \\ b\right)f(t-b)u(t-b) \\ c)f(t\left)u\right(t-a\left) \\ d\right)f\left(t\right)-f\left(t\right)u(t-b) \\ e)f(t\left)u\right(t-a)-f(t\left)u\right(t-b\left) \\ f\right)f(t-a)u(t-a)-f(t-a)u(t-b) \end{gathered}$$

Figure graph for problem 52

suppose f(t) is a function for which f’(t) is piecewise continuous and exponential order c. use result in this section and section

Justify$\mathit{f}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\underset{\mathbf{x}\mathbf{\to }\mathbf{\infty }}{\mathbf{l}\mathbf{i}\mathbf{m}}\mathit{s}\mathit{F}\mathbf{\left(}\mathit{s}\mathbf{\right)}$

Where $\mathit{F}\mathbf{\left(}\mathit{s}\mathbf{\right)}\mathbf{=}\mathit{L}\left\{f\left(t\right)\right\}$verify this result with$\mathit{f}\mathbf{\left(}\mathit{t}\mathbf{\right)}\mathbf{=}\mathit{c}\mathit{o}\mathit{s}\mathit{k}\mathit{t}\mathbf{.}$

Use Theorem 7.4.3 to find the Laplace transform of the given periodic function.

In Problems 40–54, match the given graph with one of the functions in (a)-(f). The graph is given as Figure 7.3.11.

$$\begin{gathered} a)f(t)-f(t\left)u\right(t-a\left) \\ b\right)f(t-b)u(t-b) \\ c)f(t\left)u\right(t-a\left) \\ d\right)f\left(t\right)-f\left(t\right)u(t-b) \\ e)f(t\left)u\right(t-a)-f(t\left)u\right(t-b\left) \\ f\right)f(t-a)u(t-a)-f(t-a)u(t-b) \end{gathered}$$

Figure graph for problem 53

Use Theorem 7.4.3 to find the Laplace transform of the given periodic function.

In problems 49–54, they match the given graph with one of the functions in (a)-(f). The graph of f(t) is given in figure 7. 3.11.

$$\begin{gathered} \left(a\right)f\left(t\right)-f\left(t\right)\mathcal{U}(t-a) \\ \left(b\right)f(t-b)\mathcal{U}(t-b) \\ \left(c\right)f\left(t\right)\mathcal{U}(t-a) \\ \left(d\right)f\left(t\right)-f\left(t\right)\mathcal{U}(t-b) \\ \left(e\right)f\left(t\right)\mathcal{U}(t-a)-f\left(t\right)\mathcal{U}(t-b) \\ \left(f\right)f(t-a)\mathcal{U}(t-a)-f(t-a)\mathcal{U}(t-b) \end{gathered}$$

Figure 7. 3.11. Graph for problems 49-54.

Figure 7. 3.17. Graph for problems 54.

Use Theorem 7.4.3 to find the Laplace transform of the given periodic function.

In problems, 55-62write each function in terms of unit step functions. Find the Laplace transform of the given function.

$f\left(t\right)=\left\{\begin{array}{l}2\\ -2\end{array}\right.\left.\begin{array}{r}0⩽t<3\\ t⩾3\end{array}\right\}$

Use Theorem 7.4.3 to find the Laplace transform of the given periodic function.