This is a continuation of my previous unanswered question Best method to solve this PDE
I understand that the method of characteristics, cannot be used for more than one dependent variable from here http://en.wikipedia.org/wiki/Method_of_characteristics, that is
$$ a(x,y)*\frac{\partial u}{\partial x} + b(x,y)*\frac{\partial u}{\partial y} + c(x,y)*u(x,y) = 0$$
$y$ is independent of $x$
However when we write $$\frac{dx}{da(x,y)} = \frac{dy}{db(x,y)}$$ , doesn't it imply that $y$ is dependent on $x$, i.e y $y$ can be written as a function of $x$
