In this video, the final step in solving the quasilinear PDE is to write $c_1 = f(c_2)$. The example used is the Burger Equation $uu_x + u_y = 0$, and the solution is $u = f(x - uy)$.
However, in this PDF, the final step of the same technique is to write $u = f(c_1, c_2)$. This does not seem equivalent to me. Applying the above example here we would get $u = f(u, x - uy)$, which does not seem the same as $u = f(x - uy)$.
Are these the same somehow? Is there a way to convert an arbitrary function of multiple variables into an arbitrary function of one variable? If not, which solution technique is correct?
As a secondary matter of inquiry, does it matter that this solution is often a multifunction, or is it implied that only choices of $f$ which make $u$ a function are acceptable solutions?