Solution - Two Batteries Circuit

Consider only one battery at the circuit. Because the circuit is symmetric about the middle horizontal line, we can say the circuit can be drawn as picture below.Simplify the circuit, then we can get the total resistance at the circuit, it is

Hence the voltage at the circuit is

Now, we need to find the current flow through "2R" resistors, DA and BC.
For "2R" resistor at DA, we can simply write the equation

Hence,

And the current flow through "R-R" resistors at DA is the same with above. So, the total current through DA is twice of above.

The rest current will flow through DCBA. And the current flow through "2R" resistor at BC is simply half of it, because "R-R" resistors and "2R" resistor at BC are identical.

Now consider there are two batteries connected to the circuit. Assume each of the batteries gives current to the circuit.By using superposition of two batteries, we can get the current flow through "2R" resistor at DA is

Hence the voltage through DA is

From an equation above which is relating and , we can get

Therefore,

That's my solution.
If you have any correction, question, or comment, please leave a comment below.
Thank you.