Chapter 7.6, Problem 21P

(a)

To determine

To show: The Eigen values of the given coefficient matrix are real and equal if $L>4R^{2}C$ and different if $L<4R^{2}C$.

(b)

To determine

The general solution of the system $\frac{d}{dt}\left(\begin{array}{c}I\\ V\end{array}\right)=\left(\begin{array}{cc}0& \frac{1}{L}\\ -\frac{1}{C}& -\frac{1}{RC}\end{array}\right)\left(\begin{array}{c}I\\ V\end{array}\right)$ if $R=1\Omega ,C=\frac{1}{2}F\text{ and }L=1H$.

(c)

To determine

The values $I\left(t\right)\text{ and }V\left(t\right)$ given that $I\left(0\right)=2A\text{ and }V\left(0\right)=1V$.

(d)

To determine

The limiting values of $I\left(t\right)\text{ and }V\left(t\right)$ as $t\to \infty$ and to check whether the limiting values depend on the initial conditions.