Solve for $u \equiv u(x,y)$ $$-y u_x+xu_y=0$$
I have to use characteristic method and I have a little problem with the solution proposed. My attempt starts considering auxiliary system $$\frac{dx}{ds}=-y,\frac{dy}{ds}=x \label{eq_1}\tag{1}$$
Translated with separable variable ODE $$\frac{dy}{dx}=\frac{-y}{x}\label{eq_2}\tag{2}$$
And I found the solution $$y = \frac{c}{x}\label{eq_3}\tag{3}$$
Here troubles begin because while the proposed solution follow exactly $\eqref{eq_1},\eqref{eq_2}$, then the solution of $\eqref{eq_2}$ is said to be $$x^2+y^2=c$$
Am I missing something? I've been on this quite a bit but I could not figure out what is wrong and if solutions are equivalent. Thank you very much!
When you divide the two equations in (1), you get $dy/dx = x/(-y)$, not $(-y)/x$.