math - Constant power discharge of a non-ideal capacitor

My employer sells boost converters to hold up motor drives during power loss. These boost converters are fed from capacitor banks. In order to size these banks correctly, we need to take their voltage, capacitance, and ESR into account, to ensure that there is enough available energy from the capacitors to hold up the drives for a spec'd time at a spec'd power. Right now we do this with an approximation method, but it would be nice to have a more exact equation.

We're assuming ESR, capacitance, and load power are constant.

$$ I \text{: current}\\ P \text{: power}\\ R_{C} \text{: ESR}\\ C \text{: capacitance}\\ t \text{: time}\\ V \text{: capacitor voltage}\\ \text{Standard capacitor equation:}\\I(t)=CV'(t)\\ \text{Power out of the cap equals power into the ESR plus power into the load:}\\ V(t)I(t) = P + R_{C}I^{2}(t)\\ \text{Substitute:}\\ CV(t)V'(t) = P + R_{C}C^{2}(V'(t))^{2}\\ $$

If I'm right, this gives me a non-linear differential equation, which puts me well past my mathematical comfort zone. If I understand correctly, solving a new non-linear differential equation would qualify as a significant contribution to the field of mathematical knowledge. Given that, I'm unlikely to solve this on my own.

Does anyone know any good approaches for solving for V(t)? Does anyone know if this equation has already been solved? Am I possibly misunderstanding the problem? Or should I move this to the math Stack Exchange?

Answer

The equations were solved by others here. Unless I've missed a sign somewhere, this formula gives the time it takes for a cap to reach internal voltage V, starting from voltage \$V_{0}\$, with a given ESR and capacitance, and a fixed power discharge.

$$t(V) = \frac{C}{4P} (V_{0}^2-V^2 + V_{0}\sqrt{V_{0}^2-4PR_C} - V\sqrt{V^2-4PR_C}) + CR_C(\ln\big(V+\sqrt{V^2-4PR_C}\big) - \ln\big(V_{0}+\sqrt{V_{0}^2-4PR_C}\big))$$

Note that since V is the internal, unloaded voltage of the cap, "behind" the ESR, to find the time it takes for the cap to reach a specific terminal voltage while loaded, we must use the substitution: $$V=V_{min}+\frac{PR_{C}}{V_{min}}$$ where \$V_{min}\$ is the minimum desirable terminal voltage.

These calculations appear to match our numerical estimation methods nicely.