The solution of the differential equation $x d x+y d y+\frac{x d y-y d x}{x^2+y^2}=0$, is

$y=x \tan \left(\frac{C-x^2-y^2}{2}\right)$

We have,

$x d x+y d y+\frac{x d y-y d x}{x^2+y^2}=0$

$\Rightarrow \frac{1}{2} d\left(x^2+y^2\right)+d\left(\tan ^{-1} \frac{y}{x}\right)=0$

On integrating, we obtain

$\frac{1}{2}\left(x^2+y^2\right)+\tan ^{-1} \frac{y}{x}=\frac{C}{2}$

$\Rightarrow \frac{C-x^2-y^2}{2}=\tan ^{-1} \frac{y}{x} \Rightarrow y=x \tan \left(\frac{C-x^2-y^2}{2}\right)$