Solve y'=y^2 subject to y(0)=yo.

So I need to solve y'=y^2 subject to y(0)=yo (ynaught)

So I know I can separate and get y=1/(C-x) or y=1/((1/yo)-x) after applying the condition.

My question is, why can't I just integrate to get y=y^3/3+C? Why do I have to separate?

The next question is to solve A'(t)=6sqrt(A)(9-A) with A(0)=1, and I can't figure out how I would separate this one because I'd get dA/(6sqrt(A)(9-A))=dt, but the LHS is impossible to integrate simply. So can I just take integral 6sqrt(A)(9-A) to do this?