Chapter 3.5, Problem 13P

Consider again the electric circuit in Problem 22 of Section $3.4$. This circuit is described by the system of differential equations $\frac{d}{dt}\left(\begin{array}{l}i\\ v\end{array}\right)=\left(\begin{array}{cc}0& 1/L\\ -1/C& -1/RC\end{array}\right)\left(\begin{array}{l}i\\ v\end{array}\right)$.

a) Show that the eigenvalues are real and equal if $L=4R^{2}C$.

b) Suppose that $R=1$ ohm, $C=1$ farad and $L=4$ henries. Suppose also that, $i\left(0\right)=1$ ampere and $v\left(0\right)=2$ volts. Find $i\left(t\right)$ and $v\left(t\right)$.

The circuit in Problem $22$.