Easy way to find the streamlines

Question

In a textbook, this problem appears:

Find the streamlines of the vector field $\mathbf{F}=(x^2+y^2)^{-1}(-y\hat{x}+x\hat{y})$.

The system we need to solve, I suppose, is:

$\dfrac{dx}{d\tau}=\dfrac{-y}{x^2+y^2}$

$\dfrac{dy}{d\tau}=\dfrac{x}{x^2+y^2}$

This is a text which just introduced the concept streamlines. It's not about differential equations. But I cannot find a simple way to tell what the streamlines are. The answer is "horizontal circles with the center on the $z$ axis."

I visualized this using Mathematica to confirm the answer, I also solved the system using Mathematica but the answer was very complex. I don't see how I could find that solution by hand.

Is there some trick I can use to solve this easily?

Answer 1

Hint: divide side by side the two equations (for example the second by the first), obtaing$$\frac {dy}{dx}=-\frac x{y}$$The solutions are$$x^2+y^2=c \quad (c>0)$$

Answer 2

A non-calculus answer. The dot product with the radius vector $\mathbf{r}=(x,y)$ is zero, so the streamlines are always perpendicular to the radius, and hence form circles.