<s>[INST] 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?
 [/INST] When you multiplied by $x^2 + y^2$, you transformed an exact differential equation to an inexact one. 
If we rearrange the original equation, we get
$$\tag{*}\left(x + \frac{y}{x^2 + y^2}\right)\, dx + \left(y - \frac{x}{x^2 + y^2}\right)\, dy = 0$$
Now
$$\frac{\partial}{\partial y}\left(x + \frac{y}{x^2 + y^2}\right) = 0 +\frac{x^2 - y^2}{(x^2 + y^2)^2} = \frac{x^2 - y^2}{(x^2 + y^2)^2}$$
and
$$\frac{\partial}{\partial x}\left(y - \frac{x}{x^2 + y^2}\right) = 0-\frac{y^2 - x^2}{(x^2 + y^2)^2} = \frac{x^2 - y^2}{(x^2 + y^2)^2},$$
so $(*)$ is exact and there is an $F$ such that $F_x = x + \frac{y}{x^2 + y^2}$ and $F_y = y - \frac{x}{x^2 + y^2}$. Integrating the equation $F_x = x + \frac{y}{x^2 + y^2}$ with respect to $x$, we get $F(x,y) = \frac{x^2}{2} + \arctan(x/y) + \phi(y)$ for some function $\phi(y)$ depending only on $y$. Differentiating the latter equation with respect to $y$ yields
$$F_y = -\frac{x}{x^2 + y^2} + \phi'(y) = y - \frac{x}{x^2 + y^2}.$$
Thus $\phi'(y) = y$ $\implies$ $\phi(y) = \frac{y^2}{2} + c$, where $c$ is a constant. Hence $F(x,y) = \frac{x^2+y^2}{2} + \arctan(x/y) + c$, and the general solution is
$$\frac{x^2 + y^2}{2} + \arctan(x/y) = C.$$
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