That is in the form of a solution to a first order PDE with constant coefficients:
in which you let and and then express the PDE in u in terms of v. If I then solve the equation:
using this method, I get or where is an arbitrary function of that is, the function solves my PDE like , , etc.
My favorite Basic PDE book is "Basic Partial Differential Equations" by Bleecker and Csordas which goes over this subject nicely I think. It's an easy read and I recommend it if you're studying the subject.