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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.
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