Constructing Field Lines for a Vector Field with Inverse Distance Dependence

[Incand]

Homework Statement

Determine the field lines of the vector field $\vec F = (x^2+y^2)^{-1} (-y\hat x + x\hat y)$

Homework Equations

Definition of a field line
$\frac{d\vec r (\tau ) }{d\tau } = C \vec F (\vec r (\tau ))$

The Attempt at a Solution

From the above equation and choosing $C=1$ we get
\begin{cases}\frac{dx}{d\tau } = \frac{-y}{x^2+y^2}\\
\frac{dy}{d\tau } = \frac{x}{x^2+y^2}.
\end{cases}
Separation of variables yields
\begin{cases}
\frac{x^3}{3}+y^2x = -y\tau + c_1\\
\frac{y^3}{3} + x^2y = x\tau + c_2
\end{cases}
somehow this is supposed to tell me the field lines are "horizontal circles centered around the z-axis" but I don't see how. If i multiply the first equation by x and the second by y and substract i end up with
$\frac{y^4-x^4}{3} = 2xy\tau + c_2y-c_1x$
But i still don't see why that is a circle.

 

[mfb]

You can calculate dx/dy. That gives easier formulas.

 

[Incand]

You can calculate dx/dy. That gives easier formulas.

Thanks got it now!