Let $u$ be an algebraic solution of $y'=(1/x)(y^2 + y^3)$ other than $-1$ and $0$ over $\Bbb{C}(x)$ (the quotient field of $\Bbb{C}[x]$).
So, $u$ is some fractional power series. Suppose, we adjoin $u'$ to $\Bbb{C}(x)$ to get a bigger field. Does this make sense? And if it does,then will $u$ be contained in this new field?