91Ó°ÊÓ

Suppose \(f(t)\) is a function for which \(f^{\prime}(t)\) is piece wise continuous and of exponential order \(c\). Use results in this section and Section \(4.1\) to justify $$ f(0)=\lim _{s \rightarrow \infty} s F(s), $$ where \(F(s)=\mathscr{L}\\{f(t)\\}\). Verify this result with \(f(t)=\cos k t\).

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

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

Use the Laplace transform to solve the given integral equation or integrodifferential equation. $$ \frac{d y}{d t}+6 y(t)+9 \int_{0}^{t} y(\tau) d \tau=1, \quad y(0)=0 $$

Make up a function \(F(t)\) that is of exponential order, but \(f(t) \quad F^{\prime}(t)\) is not of exponential order. Make up a function \(f(t)\) that is not of exponential order, but whose Laplace transform exists.

Solve equation (10) subject to \(i(0)=0\) with \(L, R, C\), and \(E(t)\) as given. Use a graphing utility to graph the solution for \(0 \leq t \leq 3\) $$ \begin{aligned} &L=0.1 \mathrm{~h}, R=3 \Omega, C=0.05 \mathrm{f}\\\ &E(t)=100[q u(t-1)-q u(t-2)] \end{aligned} $$

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

Find either \(F(s)\) or \(f(t)\), as indicated. $$ \mathscr{L}^{-1}\left\\{\frac{e^{-s}}{s(s+1)}\right\\} $$

Suppose that \(\mathscr{L}\left\\{f_{1}(t)\right\\} \quad F_{1}(s)\) for \(s>c_{1}\) and that \(\mathscr{L}\left\\{f_{2}(t)\right\\} \quad F_{2}(s)\) for \(s>c_{2}\). When does \(\mathscr{L}\left\\{f_{1}(t)+f_{2}(t)\right\\}\) \(F_{1}(s)+F_{2}(s) ?\)

Solve equation (10) subject to \(i(0)=0\) with \(L, R, C\), and \(E(t)\) as given. Use a graphing utility to graph the solution for \(0 \leq t \leq 3\). $$ \begin{aligned} &L=0.005 \mathrm{~h}, R=1 \Omega, C=0.02 \mathrm{f} \\ &E(t)=100[t-(t-1) ?(t-1)] \end{aligned} $$