@VictorIn_Pacific Impossibility... I do not think it means what you think it means. Mathematical impossibility simply means that the negation of a given statement is true under certain axioms.
You were looking for a function that gives back the strength of a unit. What I have shown is that you were asking the wrong question. You cannot evaluate the value of a unit, not even that of an army.
Linear formulas will also break down. If the value of a unit is scalar1 * hp + scalar2 * power, then the unit will, under its lifetime, kill expected_combat_rounds_survived * power / dicesides hitpoints, so you can write the equation expected_combat_rounds_survived * power / dicesides * scalar1 = scalar2 * power. But expected_combat_rounds_survived depends both on the average power and the deviation of power of ALL units on the board, and so does the ratio of scalar1 and scalar2.
In short, if all units have 1 power, then 1 power is worth X utilons. If all units have 2 power, then 1 power is worth X/2 utilons.
As for the differential equations for which I admittedly do not have a great affinity, my attempts at giving any sort of useful general solution failed miserably, so did those of others I asked about it. The simplified case where every unit has the same power is clearly solvable, but uninteresting, so I decided not to bother.
Anyway... I really shouldn't be spending time writing comments here. Or if I focused my attention on TripleA, I could at least get back to working on my map. So... good luck with those nasty differential equations.