When the RHS is 0, you are basically asking about the method of Riemann invariants. A quick summary of the method you can find in the second section of 

[Lax (1964), J. Math. Phys.](https://mathscinet.ams.org/mathscinet-getitem?mr=165243)

Alternatively, I am pretty sure it is also discussed somewhere in the second volume of Courant and Hilbert. 

EDIT: I see that there's also a decent write-up on [Wikipedia](https://en.wikipedia.org/wiki/Riemann_invariant), you just need to know where to look.