solve differential equation (separable)

Question

I'm solving the following differential equation: $$x+xy+y'(y+xy)=0$$ So far I've tried to rewrite it somehow to separate variables but it didn't really work. I've got a remark to introduce a substitution of some kind but couldn't think of it right away.

How do you conclude what a substitution should be in such cases?

Here's equivalent form of the eq. above: $$(1+y)xdx+(1+x)ydy=0$$

Answer 1

I don't think you really need to do a substitution. With the final form you have there, you can simply divide by $(1+x)$ and $(1+y)$ to obtain:

$$ \frac{x}{1+x} dx + \frac{y}{1+y} dy = 0 $$

Now, your equation is "separated" and you can just integrate to obtain an expression relating $y$ and $x$.

Answer 2

it is $$\frac{y}{1+y}dy=-\frac{x}{1+x}dx$$