How can I solve linear PDE with rational coefficients?

I haven't solved PDE manually in quite some time, but for my own personal projects, I'm trying to solve a two-variable PDE and then hopefully generalize it later.

Unfortunately, three different AI chatbots gave me three different answers, which is troubling. Additionally, I'm having trouble finding education on how to solve the kind of equation I'm looking for on google and youtube too, so now I thought to ask here.

The linear PDE I'm solving is

P(x,y)*U_x - Q(x,y)*U_y = 0. However, I'm starting out with a more specific and useful PDE of the form

x/(x+1)*U_x - y/(y+1)*U_y = 0. How can I find the general solution to this PDE? Because I can't double-check my work against AI, I'm not going to be 100% confident in what I write down anyway, so I'd appreciate guidance on what the solution should be and how I can reliably compute it. The method of characteristics is reliable, but there's always some small minutia about it I don't quite grasp about inferring inverse-functions and translations.