Chapter 7: The Laplace Transform

In Problems 43-46 use Problems 41 and 42 and the fact that $\Gamma \left(\frac{1}{2}\right)=\sqrt{\pi }$to find the Laplace transform of the given function.

45.$f\left(t\right)=t^{3/2}$

In Problems $37-48$find either $F\left(s\right)$or $f\left(t\right)$, as indicated $L^{-1}\left\{\frac{e^{-\pi s}}{s^2+1}\right\}$.

Use the Laplace Transform and these inverses to solve the given initial-value problem

${\mathit{y}}^{\mathbf{\text{'}}}\mathbf{+}\mathit{y}\mathbf{=}{\mathit{e}}^{\mathbf{-}\mathbf{3}\mathbf{t}}\mathit{c}\mathit{o}\mathit{s}\mathbf{2}\mathit{t}\mathbf{,}\mathit{y}\left(0\right)\mathbf{=}\mathbf{0}$

The current i(t) in an RC-series circuit can be determined from the integral equation.

$\mathbf{R}\mathbf{i}\mathbf{+}\frac{\mathbf{1}}{\mathbf{C}}\underset{\mathbf{0}}{\overset{\mathbf{t}}{\mathbf{\int }}}\mathbf{i}\left(\tau \right)\mathbf{d}\mathbf{\tau }\mathbf{=}\mathbf{E}\mathbf{\left(}\mathbf{t}\mathbf{\right)}$,

where E(t) is the impressed voltage. Determine when

$\mathit{R}\mathbf{=}\mathbf{10}\mathit{\Omega }\mathbf{,}\mathit{C}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{5}\mathit{f}\mathbf{,}$and$\mathbf{E}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}\mathbf{2}\mathbf{(}{\mathbf{t}}^{\mathbf{2}}\mathbf{+}\mathbf{t}\mathbf{)}$

In Problems 43-46 use Problems 41 and 42 and the fact that$\mathit{\Gamma }\left(\frac{1}{2}\right)\mathbf{=}\sqrt{\mathbf{\pi }}$ to find the Laplace transform of the given function.

46.$f\left(t\right)=2t^{1/2}+8t^{5/2}$

In problem and use the Laplace transform and these inverse to solve the given initial-value problem.

${\mathit{y}}^{\mathbf{n}}\mathbf{-}\mathbf{2}\mathit{y}\mathbf{\text{'}}\mathbf{+}\mathbf{5}\mathit{y}\mathbf{=}\mathbf{0}\mathbf{,}\mathit{y}\left(0\right)\mathbf{=}\mathbf{1}\mathbf{,}\mathit{y}\mathbf{\text{'}}\left(0\right)\mathbf{=}\mathbf{3}$

In Problems$37-48$find either$F\left(s\right)$or$f\left(t\right)$, as indicated $L^{-1}\left\{\frac{s{e}^{-\pi s/2}}{s^2+4}\right\}$.

A series circuit contains an inductor, a resistor, and a capacitor for which $L=\frac{1}{2}h,R=10\Omega ,$and $C=0.01f$, respectively. The voltage

$$\begin{gathered} \mathbf{E}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}\left\{10,0⩽t<5 \\ 0,t⩾5\right\} \end{gathered}$$

is applied to the circuit. Determine the instantaneous charge q(t) on the capacitor for t>0 if q(0)=0 and q'(0)=0

Suppose that$\mathcal{L}\left\{f_1\left(t\right)\right\}=F_1\left(s\right)$ for $s>c_1$and that$\mathcal{L}\left\{f_2\left(t\right)\right\}=F_2\left(s\right)$ for $s>c_2$. When does

$\mathcal{L}\left\{f_1\left(t\right)+f_2\left(t\right)\right\}=F_1\left(s\right)+F_2\left(s\right)?$

In problems, $37-48$find either $F\left(s\right)$or$f\left(t\right)$, as indicated $L^{-1}\left\{\frac{e^{-s}}{s(s+1)}\right\}$.