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?

In terms of polar coordinate $(x,y) = (r\cos\theta,r\sin\theta)$, we have
$$\begin{align}
xdx + ydy 
&= 
r\cos\theta(\cos\theta dr - r\sin\theta d\theta) + 
r\sin\theta(\sin\theta dr + r\cos\theta d\theta) = r dr\\
\frac{xdy - ydx}{x^2+y^2} 
&=
\frac{r\cos\theta(\sin\theta dr + r\cos\theta d\theta) - r\sin\theta(\cos\theta dr - r\sin\theta d\theta)}{r^2} = d\theta
\end{align}$$
This leads to
$$dr^2 = 2rdr = 2d\theta
\;\;\implies\;\;
r^2 = 2\theta + C
\;\;\iff\;\;
x^2 + y^2 = 2\tan^{-1}\left(\frac{y}{x}\right) + C
$$
for suitably chosen constant $C$.
