Chapter 7: The Laplace Transform

The Laplace transform $\mathbf{L}\left\{{\mathbf{e}}^{\mathbf{-}{\mathbf{t}}^{\mathbf{2}}}\right\}$exists, but without finding it solve the initial-value problem$\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{y}\mathbf{=}{\mathbf{e}}^{\mathbf{-}{\mathbf{t}}^{\mathbf{2}}}$,$\mathbf{y}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{0}$,$\mathbf{y}\mathbf{\text{'}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{0}$.

In Problems 63–70, use the Laplace transform to solve the given initial-value problem.

$y+4y=f\left(t\right),y\left(0\right)=0,y\left(0\right)=-1wherewheref\left(t\right)=\left\{\begin{array}{ll}1,& 0⩽t<1\\ 0,& t⩾1\end{array}\right.$

Solve the integral equation $\mathbf{f}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}{\mathbf{e}}^{\mathbf{t}}\mathbf{+}{\mathbf{e}}^{\mathbf{t}}{\mathbf{\int }}_{\mathbf{0}}^{\mathbf{t}}{\mathbf{e}}^{\mathbf{-}\mathbf{\tau }}\mathbf{f}\mathbf{\left(}\mathit{\tau }\mathbf{\right)}\mathit{d}\mathit{\tau }$.

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

$y\text{'}\text{'}+4y=\sin t\mathcal{U}(t-2\pi ),y\left(0\right)=1,y\text{'}\left(0\right)=0$

(a) Show that the square wave function $E\left(t\right)$given in Figure 7.4.4 can be written $\mathit{E}\mathbf{\left(}\mathit{t}\mathbf{\right)}\mathbf{}\mathbf{=}\mathbf{}\mathbf{}\mathbf{\sum }_{\mathbf{k}\mathbf{=}\mathbf{0}}^{\mathbf{x}}{\mathbf{(}\mathbf{}\mathbf{-}\mathbf{}\mathbf{1}\mathbf{)}}^{\mathbf{k}}\mathit{U}\mathbf{(}\mathit{t}\mathbf{}\mathbf{-}\mathbf{}\mathit{k}\mathbf{)}$.

(b) Obtain (14) of this section by taking the Laplace transform of each term in the series in part (a).

In Problems 63-70 use the Laplace transform to solve the given initial-value problem.

$y\text{'}\text{'}-5y\text{'}+6y=\mathcal{U}(t-1),y\left(0\right)=0,y\text{'}\left(0\right)=1$

. Use the Laplace transform as an aide in evaluating the improper integral

${\mathbf{\int }}_{\mathbf{0}}^{\mathbf{x}}{\mathbf{te}}^{\mathbf{-}\mathbf{2}\mathbf{t}}\mathit{s}\mathit{i}\mathit{n}\mathbf{4}\mathit{t}\mathit{d}\mathit{t}$

In problems, 63-70 use the Laplace transform to solve the given initial-value problem.

$y\text{'}\text{'}+y=f\left(t\right),y\left(0\right)=0,y\text{'}\left(0\right)=1wheref\left(t\right)=\left\{\begin{array}{ll}0,& 0⩽t<\pi \\ 1,& \pi ⩽t<2\pi \\ 0,& t⩾2\pi \end{array}\right.$

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

6.$f\left(t\right)={\{}_{\cos t}^{0,}\raisebox{1ex}{$\raisebox{1ex}{$0\le t\pi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.$}\!\left/ \!\raisebox{-1ex}{$\raisebox{1ex}{$t\ge \pi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.$}\right.$

In Problems 1–12 use the Laplace transform to solve the given initial value problem.

$\mathit{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathit{y}\mathbf{=}\mathit{\delta }\mathbf{(}\mathit{t}\mathbf{-}\mathbf{2}\mathit{\pi }\mathbf{)}\mathbf{+}\mathit{\delta }\mathbf{(}\mathit{t}\mathbf{-}\mathbf{4}\mathit{\pi }\mathbf{)}\mathbf{,}\mathbf{}\mathit{y}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{1}\mathbf{,}\mathbf{}\mathit{y}\mathbf{\text{'}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{0}$