$(50$ pts$)$ Consider the electrical circuit below. $v_1(t)$ and $v_2(t)$ are batteries with the polarities shown. $v_3(t)$ is the voltage drop across the capacitor to ground. Ground may be attached to bottom wire. Determine the differential equations of motion for $i_1(t)$ and $i_2(t)$ in standard form. Draw a block diagram of the system with the batteries as inputs and the currents as outputs.