10. Write two differential equations for the system by using
cyclic currents, beginning at the instant the switch is
closed. The dependent variables should be the charges.
Express the equations in a state-space representation,
and use software to find the damped natural frequencies,
solve, and plot the current through the resistor vs. time.
Use $e(t) = 12\sin(2t)$ volts, $C_1 = 1000\mu F$, $C_2 = 2000\mu F$, $R = 50\Omega$,
$L_1 = 8H$, and $L_2 = 3H$, assume zero initial conditions.