A first order linear PDE of the form $$a(x,y)u_{x}+b(x,y)u_{y}=c(x,y)$$ can be solved (under some conditions) using the method of characteristics. This method transforms the PDE above into two ODE's, namely $$ \frac{dx}{dy}=\frac{a}{b}\\ ~\frac{du}{dy}=\frac{c}{b}.$$
Bearing in mind that $u$ is a function of two variables, $x$ and $y$, what does $\dfrac{du}{dy}$ mean in the second ODE? Put differently, do we consider $u$ to be a function of $y$ only in the second ODE? Isn't this inconsistent since we know that $u$ also depends on $x$?