I'm doing some practice to solve any exact(not) differential equations and i found this one. I'm having difficulty to determine the integrating factor which is really troublesome. I can't figure out the integrating at all, since it's not depends on one of the form $x$, $y$, $xy$, $x^2+y^2$.

$$ \left(2x + \ln y\right) dx + \left( xy \right)dy = 0$$ My attempt to determine the integrating factor:

$M_y=\frac{1}{y}\neq N_x =y$,


$\frac{N_x-M_y}{M}=\frac{y^2-1}{y}.\frac{1}{2x+\ln y}$

$\mu(x,y)=\frac{M_y-N_x}{yN-xM}=\frac{\frac{1}{y}-y}{xy^2-x(2x+\ln y)}$

$\mu(x^2+y^2)=\frac{M_y-N_x}{2(xN-yM)}=\frac{\frac{1}{y}-y}{2(x^2y-y(2x+\ln y))}$

I have used both wolframalpha and maple, but i found no helps. Anybody could help me out of here? Thanks in advance.


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