Chapter 7: The Laplace Transform

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

$f\left(t\right)=\left\{\begin{array}{l}1,\\ 0,\\ 1,\end{array}\right.\left.\begin{array}{r}0⩽t<4\\ 4⩽t<5\\ ⩽t\ge 5\end{array}\right\}$

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

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.

$f\left(t\right)=\left\{\begin{array}{ll}0,& 0⩽t<3\pi /2\\ \sin t,& t⩾3\pi /2\end{array}\right.$

In Problems 59 solve equation (15) subject to $\mathbf{i}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{0}$with E(t) as given. Use a graphing utility to graph the solution for$\mathbf{0}⩽\mathbf{t}⩽\mathbf{4}$in the case when$\mathbf{L}\mathbf{=}\mathbf{1}$and$\mathit{R}\mathbf{=}\mathbf{1}$

E(t) is the meander function in Problem 53 with amplitude 1and a=1.

Write each function in terms of unit step functions. Find the Laplace transform of the given function.

$f\left(t\right)=\left\{\begin{array}{l}0,\\ t^2,\end{array}\right.\left.\begin{array}{r}0⩽t<1\\ t\ge 1\end{array}\right\}$

Use the Laplace transform to solve the given initial- value problem

$\mathbf{}\mathbf{5}\mathbf{.}\mathbf{}\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{y}\mathbf{=}\mathbf{\delta }\left(\mathbf{t}\mathbf{-}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{\pi }\right)\mathbf{+}\mathbf{\delta }\left(\mathbf{t}\mathbf{-}\frac{\mathbf{3}}{\mathbf{2}}\mathbf{\pi }\right),\mathbf{y}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{0} ,\mathbf{y}\mathbf{\text{'}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{0}$

In Problems 1–18 use Definition 7.1.1 to find$L\left\{f\left(t\right)\right\}.$

5.$f\left(t\right)={\{}_{1,t⩾\pi }^{\sin t,0⩽t<\pi }$

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

In Problems 59 solve equation (15) subject to$\mathbf{i}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{0}$with E(t) as given. Use a graphing utility to graph the solution for$\mathbf{0}⩽\mathbf{t}⩽\mathbf{4}$in the case when$\mathit{L}\mathbf{=}\mathbf{1}$and$\mathit{R}\mathbf{=}\mathbf{1}$

E(t) is the sawtooth function in Problem 55 with amplitude 1

and b = 1.