Considering the very simple electrical dc power flow problem with two nodes, with voltage V1 and V2 respectively, connected via a resistance R, we end up with a system of two nonlinear equations:
We have two equations and four unknowns. I would expect then that, defining two variables (let's say $P_1$ and $P_2$), the system would be solvable for the other two.
However, when using $P_1 = P_2 = 0$, I get infinite solutions where $V_1 = V_2$.
I got curious about this. Is there any property/parameter of nonlinear equation systems that says anything about its solvability/number of solutions?
More specifically, when solving those kind of systems, how do I know the conditions where there will be infinite solutions instead of one or two?