How to prove that a curve is characteristic? (PDE's)

Question

I'm having problems proving that a curve is characteristic, in PDE's

Assume that $r(t)$ is a parametrized curve on XY plane. If $F(r(t)) = r'(t)$, we say that that parametrized curve is a characteristic curve.

Now, I don't know that $r(t)$ or $F(r(t))$ are…

I have an example for you:

$xUy - yUx = 0$

And I must find the solution that contains the curve $x = 0$; $U = y^2$

Altough I know how to find a solution, it says here that the curve mentionated earlier is characteristic and therefore it has an infinite amount of solutions.

So my questions are:

• Why does a characteristic curve have infinite solutions in an ODE? (maybe I'm asking that because I don't know very well the geometric meaning of it...). And why does a non-characteristic curve only has 1 solution?

• How can I prove that in the concrete example that I mentionated?

EDIT: I'm editing some stuff because my previous question may not be very clear.

Answer 1

As $dU = U_x dx+U_y dy=0$ we have

$$ \frac{dy}{dx}=-\frac{U_x}{U_y} = -\frac xy $$

so

$$ y dy + x dx = 0 $$

is the equation of a generalized characteristic curve or

$$ x^2+y^2= C $$

NOTE

The constant $C$ characterizes a parametric family of circles (infinite possibilities).