What is the tip for this exact differential equation? $$ xdx + ydy = \frac{xdy - ydx}{x^2 + y^2} $$
I have multiplied the left part $x^2+y^2$ for $x dx + y dy$ getting 
$$(x^3+xy^2+y)dx+(x^2y+y^3-x)dy=0$$ 
And the derivative test give me: 
$\frac{dM}{dy}= 0+2xy+1$ and $\frac{dN}{dx} = 2xy+0-1$. 
Where´s my mistake?

$\frac{1}{2}d(x^2+y^2) = xdx+ydy$
$d(y/x) = \frac{xdy-ydx}{x^2}$
Let $u = x^2+y^2$ and $v = \frac{y}{x}$ 
$LHS = \frac{1}{2}d(x^2+y^2) = \frac{1}{2}du$
$RHS = \frac{xdy-ydx}{x^2+y^2} = x^2 \frac{dv}{u} = \frac{u}{1+v^2}\frac{dv}{u} = \frac{dv}{1+v^2}$
Thus,
$$
\frac{du}{dv} = \frac{1}{1+v^2}\
$$
Now integrate and substitute for $u,v$
