Solutions of $~\frac{\partial{u}}{\partial{x}} = 0$: As the solutions don't depend on $x$, but constant on the lines $y =$ constant in the $xy$ plane My PDE textbook says the following:

The simplest possible PDE is $~\frac{\partial{u}}{\partial{x}}  = 0$ [where $u = u(x, y)$]. Its general solution is $u = f(y)$, where $f$ is any function of one variable. For instance, $u = y^2 - y$ and $u = e^{y} \cos(y)$ are two solutions. Because the solutions don't depend on $x$, they are constant on the lines $y =$ constant in the $xy$ plane.

I'm struggling to visualize and understand what the author means by this last part:

Because the solutions don't depend on $x$, they are constant on the lines $y =$ constant in the $xy$ plane.

I would appreciate it if someone could please take the time to clarify this.
 A: It would be best if you thought in three dimensions, so you have a normal x-y Cartesian plane below you.  Now imagine what the three dimensional surface, say, $z = u(x,y) = x^2+y^2$  That surface would look like a parabola 'rotated' around the line where (x,y)=(0,0).  
I agree the explanation is confusing, but what the author is saying is that any function of y alone will always have $\partial u/\partial x=0$ since there is no x involved.  
But if you think about any of those functions, they all share that the value does not depend on x.  It does depend on y, so it might be a parabola (that is, the z coordiante) as you move along the y axis up and down.  But, since it does not depend at all on x, the value must be the same for all values of x.  
In other words, start with any x you like (let's say 0) and pick any y and there is a corresponding value that gives you the height along the z axis.  But, if you now move left or right along the x axis, the value of z can't change - since the partial derivative with respect to x is zero.  
$u=y^2$ is probably easiest to visualize.  when $x=0$, it is indeed a parabola viewed along the z axis.  But, it is exactly the same parabola at every point x.  So it's sort of like you took the parabola and slid it from x equals minus infinity to infinity, and carved out that surface.  Then it should be easy to see that (for example) when y=2 then u(x,y)=4 independent of x: i.e., it is a constant for all values of x.
