I'd really love your help with solving the following differential equation: xy'-y=(x+y)(\ln(x+y)-\ln(x)).
I transformed the equation to $((x+y)(\ln x-\ln(x+y))-y)dx+xdy=0$ and then checked if $\frac{(x+y)(\ln x-\ln(x+y))-y}{dy}=\frac{x}{dy}$ in order to find $\xi$ such that $\frac{d\xi}{dx}$=$\frac{d\xi}{dy}$, but it doesn't. Then I tried to multiply the equation with both $\mu(x) $ and $\mu(y)$ to find a integration factor in order to find $\xi$ but both \frac {\mu'(x)}{\mu(x)} and \frac {\mu'(y)}{\mu(y)} were depended on both $x$ and $y$.
Any suggestion for how should I solve this one?
Thanks a lot!