Chapter 3.4, Problem 22P

Applications.

The electric circuit shown inFigure $\mathrm{3.4.6}$ is described by the system of differentil equations

$\frac{d}{dt}\left(\begin{array}{l}i\\ v\end{array}\right)=\left(\begin{array}{cc}0& \frac{1}{L}\\ -\frac{1}{C}& -\frac{1}{RC}\end{array}\right)\left(\begin{array}{l}i\\ v\end{array}\right)$, (i)

Where $i$ is the current through the inductor and $v$ is the voltage across the capacitor. These differential equations were derived in Problem $27$ of Section $3.2$.

(a) Show that the eigenvalues of the coefficient matrix are real and different if $L>4R^{2}C$ and complex conjugates if $L<4R^{2}C$.

(b) Suppose that $R=1$ ohm, $C=\frac{1}{2}$ farad, and $L=1$ henry. Find the general solution of the system (i) in this case.

(c) Find $i\left(t\right)$ and $v\left(t\right)$ if $i\left(0\right)=2$ amperes and $v\left(0\right)=1$ volt.

(d) For the circuit of part (b), determine the limiting values of $i\left(t\right)$ and $v\left(t\right)$ as $t\to \infty$. Do these limiting values depend on the initial conditions?