Chapter 3.4, Problem 21P

Applications.

Consider the electric circuit shown in Figure $\mathrm{3.4.5}$. Suppose that $R_1=R_2=4$ ohms, $C=\frac{1}{2}$ farad, and $L=8$ henries.

(a) Show that this circuit is described by the system of differential equations

$\frac{d}{dt}\left(\begin{array}{l}i\\ v\end{array}\right)=\left(\begin{array}{cc}-\frac{1}{2}& -\frac{1}{8}\\ 2& -\frac{1}{2}\end{array}\right)\left(\begin{array}{l}i\\ v\end{array}\right)$, (i)

Where $i$ is the current through the inductor and $v$ is the voltage across the capacitor.

Hint: See Problem $27$ of Section $3.2$.

(b) Find the general solution of Eqs. (i) in terms of real valued functions.

(c) Find $i\left(t\right)$ and $v\left(t\right)$ if $i\left(0\right)=2$ amperes and $v\left(0\right)=3$ volts.

(d) 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?