
Domain of an ODE
I am not understanding how the domain is supposed to work. I know that x cannot be 0, but I'm confused about the interval. Looking at the direction field for the solution that goes through (1,1), I see a parabola that seems to be bounded by x=2 and x=0. What does this mean in terms of the domain?
$\displaystyle \frac{dy}{dx}=(1+x)(1+y)^2, \;\; y(1)=1$

OK, perhaps I was being too vague? Is the domain x<0?

Alright, then I would just solve it:
$\displaystyle y(x)=\frac{2+2x+x^2}{x(2+x)}$
Bingo: $\displaystyle x\notin\{0,2\}$

Looks like I made an algebraic error, getting $\displaystyle y=\frac{2}{x^2}\frac{1}{x}1$. So the domain would be 2<x<0, then? That would make sense with regards to the direction field.