I am formulating a set of relationships between two variables, x and y. Both variables represent real world quantities, and have associated units. These units are not the same for x and for y.
My plan was to formulate a model linear (gradient) system, with a quadric potential $V(x,y)$: $$ -\frac{\partial V}{\partial x} = \frac{dx}{dt} = ax + by\\ -\frac{\partial V}{\partial y} = \frac{dy}{dt} = cx + dy $$
I believe for the potential $V$ to be well-defined, I need to require that $b=c$ (requirement for a gradient system I think?).
I am immediately confused because of the units of $b$. Using square brackets to denote, "units of", does $b$ have units of $[\frac{x}{y\cdot t}]$ as in the first equation or units of $[\frac{y}{x\cdot t}]$ as in the second?
Is this something fundamental about potential fields / gradient systems, that the variables must have the same units?