This question was answered Help with characteristic method technicality
with the assumption that $c_1 = \Phi(c_2).$
But, I'm having trouble seeing how they picked the specific form they did. What is the basis for their assumption? If you want to transform one constant into another, you can just add another constant $c_1 + a = c_2.$
It looks like it has something to do with the answer to this question Family of characteristic curves of a first-order quasi-linear pde but I'm not sure, I still don't quite understand the basis for the assumption and how it explicitly looks symbolically.
So the solution $u(x,y)$ is constant with respect to this proxy variable s. Okay, I'm sure this follows from vector calculus. I understand the concept that you propagate a paramatrically defined ODE to construct an integral surface, but still don't understand why any of that implies this functional relationship.
