I’m studying the flyback converter. I don't understand how he finds the equation in red when the MOS is turned off. I find the same equation with the -, and I don't think it got the tension with the opposite.

 

I will give it a try:
Vo is the output voltage. (5V in my example)
n is probably the turn ratio. (10:1)
The voltage across the transformer is 50V on the primary and 5V on the secondary. (when the MOSFET is off and the diode is on)
Using my numbers, the voltage at point where transformer and LLK connect is 50V above Vin.
Vsn must be the voltage across the snubber cap.

I changed your circuit. During the flyback time the voltage on the MOSFET is very high. I drew where the top of the page is high voltage and ground is at the bottom of the page.

 

I don’t understand. If we consider kirchhoff's law in the loop when the MOS is OFF, the situation should be that in the figure.