Chapter 3.3, Problem 32P

Electric Circuits. Problem $32$ and $33$ are concerned with the electric circuits described by the system of differential equations in problem $29$ of section $3.2:$

$\frac{d}{dt}\left(\begin{array}{l}i\\ v\end{array}\right)=\left(\begin{array}{cc}-R_1/L& -1/L\\ 1/C& -1/C{R}_2\end{array}\right)\left(\begin{array}{l}i\\ v\end{array}\right)$. $\left(\text{i}\right)$

(a) Find the general solution of Eq. (i) if $R_1=1\text{ohm,}{R}_2=\frac{3}{5}\text{ohm,}\text{L}\text{=}\text{2}\text{henries,}$ and $C=\frac{2}{3}\text{farad}$.

(b) Show that $i(t)\to 0$ and $v(t)\to 0$ as $t\to \infty$, regardless of the initial values $i(0)$ and $v_0$.

Section $3.2$