When we use the method of characteristics to solve a quasilinear PDE \begin{equation} a(x,y,u)u_x + b(x,y,u)u_y = c(x,y,u), \end{equation} we use the method of characteristics to find the characteristic equations defined by \begin{equation} \frac{dx}{dy} = \frac{a}{b} , \quad \frac{du}{dx} = \frac{c}{a} \end{equation} Then once we solve the above ODE's in terms of the constants $c_1$ and $c_2$, we relate the two through an arbitrary function $c_2 = f(c_1)$. I don't understand why we do this though. I've searched everywhere, but I can't find a good description explaining this step. The best I can find is along the lines of:
If you can find characteristic curves $u_1(x,y,u)=c_1$ and $u_2(x,y,u)=c_2$, then the general solution to the PDE can be written as an implicit function satisfying the equation $$F(u_1,u_2) = F(c_1,c_2) = 0$$ for some arbitrary function $F$. We can then solve for $c_2$ to recover $c_2 = f(c_1)$ for some $f$.
The problem I have with the above though, is again why we can we write the general solution to the PDE as $F(c_1,c_2) = 0$?
There is a gap in my knowledge here that I can't find the answer to anywhere, so I would very much appreciate someone's help. Thank you!