Question on PDE concepts (Introductory course) Extracted from the book on PDE written by Strauss Walter, there is a question in the very first chapter on solving PDE using the idea of THE VARIABLE COEFFICIENT EQUATION. Here, I give an example so as to state my confusion.
Solve the PDE 
$$u_x + 2xy^2u_y = 0$$
Currently, following the initial steps, I decompose the question into an ODE, where the characteristic curves are of the form 
$$y = \frac{1}{C - x^2}$$
Now, it says that the function $u(x,y)$ is a constant on each such curve because 
$$\frac{d}{dx}u \left(x,\frac{1}{C - x^2} \right) = \cdots = 0$$ 
I have trouble understanding what $u(x,y)$ is constant on each such curve means? And why is it sufficient to solve for $C$.
If my question is not clear, I will add more details.

The concept of level curves or level surfaces are used in calculus and differential equations.
A level curve for a function $u=u(x,y)$ is a curve defined by a relation between $x$ and $y$ such that $u(x,y)$ is constant on that curve. 
For example $u(x,y)=x^2+y^2$ has level curves at $x^2+y^2=C$ which are circles for positive values of C. 
$$\frac{d}{dx}u(x,(C - x^2)^{-1}) = \cdots = 0$$ speaks to the same concept because derivative of a constant is $0$.
