Analysis of a SEPIC (calculating duty cycle with coupled inductors)

Hi - I have been working on researching and understanding SEPIC power supplies. I was able to calculate duty cycle (and everything else that was interesting to me) of a SEPIC (operating in continuous mode) when the inductors are not coupled. I made the assumption that the capacitors had a constant voltage, which I think should be good enough for a first order approximation. I solved for the duty cycle by looking at the change in current of each of the inductors while the switch is on, and then looking at the change in current in each inductor when the switch is off. The sum of the changes (ie IL1on + IL1off) should be zero - so that gave me two equations with two unknowns - DC and Vc (voltage of the series capacitor). Very solvable.

But I'm just not sure where to even start when you have coupled inductors. I believe the duty cycle stays the same but the ripple is halved, but I'd like to actually see the math that shows me why that is.

Any suggestions?

Thank you!