= Solution
Near the <initial condition> $y=1$, divide by $y^2$ and separate variables: $y^{-2}\,dy=-x^2\,dx$. Integration gives $-1/y=-x^3/3+C$; the initial value fixes $C=-1$. Thus the solution of the <separable differential equation> is
$$
\boxed{y(x)=\frac1{1+x^3/3}=\frac3{x^3+3},\qquad x>-\sqrt[3]3.}
$$
The stated interval is the maximal connected interval containing zero: the solution diverges at its left endpoint. The constant solution $y=0$ does not satisfy the <initial condition>.
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