Hello i have been trying to solve the differential equation :
$$xdx+ydy=\left(\sqrt{x^2+y^2}\right)dx$$

By integrating factors i supose that:

$$\frac{\partial\mu(x,y)}{\partial y}=0$$

so that i get 

$$\frac{y\mu}{\sqrt{x^2+y^2}}=\frac{\partial\mu(x,y)}{\partial x}$$

but then solving this i get:

$$\mu=\frac{\sqrt{x^2+y^2}+x}{y}$$

and the ODE becomes:

$$-ydx+(\sqrt{x^2+y^2}+x)dy=0$$

with $$M(x,y) = -y$$ and $$N(x,y) = \sqrt{x^2+y^2}+x $$

But 
$$\frac{\partial M}{\partial y} = -1$$

$$\frac{\partial N}{\partial x} = 1+\frac{x}{\sqrt{x^2+y^2}}$$

and $$ \frac{\partial M}{\partial y} \not= \frac{\partial N}{\partial x} $$ 

so can anybody say to me where i did a mistake ?