Exact ODE:$ \left(\frac{y}{x}+6x\right)dx + (\log(x) - 2)dy =0$

Question

I need to solve the equation $$\left(\frac{y}{x}+6x\right)dx + (\log(x) - 2)dy=0$$

We can easly see that $$\left(\frac{y}{x}+6x\right)'_y =(\log(x) - 2)'_x$$

but the domain of $ A(x,y)=(\frac{y}{x}+6x, \log(x) -2)$ is not simply connected (or am I wrong?). Hpw can I show that the field A is conservative (if it is) and solve the equation? Is there any other way?

thanks!

@Edit:

Indeed, the filed is conservative. We shpuld take

$$\phi(x,y)=y \log(x)+3x^2-2y$$

Answer 1

Hint

I do not know how much this could help you; so, please forgive me if I am off-topic.

What you observed suggests (at least to me) that a change of variable $$y(x)=\frac{z(x)}{\log (x)-2}$$ could be quite interesting to explore. The differential equation becomes quite nice.