Answer updated- please check the signs
I am pretty sure I got the signs right in this answer. The point is that a current pointing in the direction of i_1 will reduce the voltage V_1, because the current takes the positive charge away from the capacitor. A current pointing in the direction of i_2 will increase the voltage V_2 for the same reason.
Finally, Lenz's law states that a potential difference in an inductor induces a current pointing in the opposite direction, so dI/dt should point in the opposite direction of the potential difference.