Select the initial-value branch after a separated integration

$y\,dy=x^2\,dx$ gives $y^2/2=x^3/3+C$. With $(x,y)=(1,1)$, $C=1/2-1/3=1/6$. Thus the source implicit relation is $y^2=2x^3/3+1/3$.

The initial y-value is positive, so the classical solution is $y=\sqrt{(2x^3+1)/3}$ on the maximal interval $x>-\sqrt[3]{1/2}$. The negative square-root branch does not pass through $(1,1)$; the endpoint y=0 is excluded because the original differential equation divides by y.