91Ó°ÊÓ

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 \(0 \leq t \leq 3\). $$\begin{aligned}&L=0.1 \mathrm{h}, R=3 \Omega, C=0.05 \mathrm{f}\\\&E(t)=100[\mathscr{U}(t-1)-\mathscr{U}(t-2)]\end{aligned}$$

Show that the function \(f(t)=1 / t^{2}\) does not possess a Laplace transform. [Hint: Write \(\mathscr{L}\left\\{1 / t^{2}\right\\}\) as two improper integrals: \\[\mathscr{L}\left\\{1 / t^{2}\right\\}=\int_{0}^{1} \frac{e^{-s t}}{t^{2}} d t+\int_{1}^{\infty} \frac{e^{-s t}}{t^{2}} d t=I_{1}+I_{2}\\] Show that \(I_{1}\) diverges.

Suppose \(f(t)\) is a function for which \(f^{\prime}(t)\) is piecewise continuous and of exponential order \(c .\) Use results in this section and Section 7.1 to justify $$f(0)=\lim _{s \rightarrow \infty} s F(s)$$ where \(F(s)=\mathscr{L}\\{f(t)\\} .\) Verify this result with \(f(t)=\cos k t.\)

If \(\mathscr{L}\\{f(t)\\}=F(s)\) and \(a>0\) is a constant, show that \\[\mathscr{L}\\{f(a t)\\}=\frac{1}{a} F\left(\frac{s}{a}\right)\\] This result is known as the change of scale theorem.

Write each function in terms of unit step functions. Find the Laplace transform of the given function. $$f(t)=\left\\{\begin{array}{lr} 2, & 0 \leq t<3 \\ -2, & t \geq 3 \end{array}\right.$$

Write each function in terms of unit step functions. Find the Laplace transform of the given function. $$f(t)=\left\\{\begin{array}{lr} 1, & 0 \leq t<4 \\ 0, & 4 \leq t<5 \\ 1, & t \geq 5 \end{array}\right.$$

Write each function in terms of unit step functions. Find the Laplace transform of the given function. $$f(t)=\left\\{\begin{array}{lr} 0, & 0 \leq t<1 \\ t^{2}, & t \geq 1 \end{array}\right.$$

Write each function in terms of unit step functions. Find the Laplace transform of the given function. $$f(t)=\left\\{\begin{array}{lr} 0, & 0 \leq t<3 \pi / 2 \\ \sin t, & t \geq 3 \pi / 2 \end{array}\right.$$

Write each function in terms of unit step functions. Find the Laplace transform of the given function. $$f(t)=\left\\{\begin{array}{lr} t, & 0 \leq t<2 \\ 0, & t \geq 2 \end{array}\right.$$

Write each function in terms of unit step functions. Find the Laplace transform of the given function. $$f(t)=\left\\{\begin{array}{lr} \sin t, & 0 \leq t<2 \pi \\ 0, & t \geq 2 \pi \end{array}\right.$$