Deriving a differential equation from a mathematical model- Mechanics sort of

[Levido]

Hi all, I have a problem deriving the differential Equation they use in the question. While not strictly asked for in the question, I still would like to know how to derive it for my own skills. I don’t need help solving the differential equation (yet).

Here’s the question:


And here’s my thinking. It’s brief because I have no idea where to go from here:


Sorry if this is the wrong section, I couldn’t see a mechanics section

 

[Levido]

If you don’t understand how to help please let me know, anything to get rid of this silence and get a conversation going. In simpler words I’m asking for help deriving the differential equation they say ”it can be shown that etc”. My problem is definitely that I’ve never met rough pulleys before but there doesn’t seem to be a good online resource about this.