Question is as simple as:
What are the different methods for solving a first-order PDE?
I'm aware of nearly all forms of Method of Characteristics - Lagrange Method, Charpit's Method.
I'm learning PDE, and was curious if there are any other methods since most of the links suggest (only) method of characteristics. If there are any, it'll be great if you can give an outline of it or a link where I can read about it, to learn!
I read through this unanswered question, I'm not considering higher dimensions, I'm talking about simpler cases.
Another method that you can employ to solve first order PDE (actually it works for other types too, but in the proofs one reduces everything to a first-order system) is the Cauchy-Kowalevski theorem. The idea here is to exploit analyticity in order to expand your solution as a power series. You can read more on wikipedia or in pretty must any basic PDE book (e.g. Chapter 4.6 in Evans).