Chapter 3.3, Problem 33P

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

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

Consider the preceding system of differential equations $\left(\text{i}\right)$.

(a) Find a condition on $R_1,R_2,C,$ and $L$ that must be satisfied if the eigenvalues of the coefficient matrix are to be real and different.

(b) If the condition found in part (a) is satisfied, show that both eigenvalues are negative. Then show that $i(t)\to 0$ and $v(t)\to 0$ as $t\to \infty$, regardless of the initial conditions.

Section $3.2$