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Is this the ODE?\[\left(\frac{1}{x+y}+2y(x+1)\right)y'-\frac{y}{x(x+y)}=-y^2\]

yes!

so the method of a total differential?

Right

You're welcome!

i can't solve it :(

How to get this? Please, explain me \[\Phi(x,y)=C\\(x+1)y^2+\log(x+y)-\log x=C\]

okay thank a lot all! :)

personally, i think this is a really dodgy/interesting DE
it totally blows up around (0,0)