 Solution

Where $y\ne 0$ and $x^2\ne 1$, the separable differential equation givesThe requested point selects the implicit solution $2y^2+1=(x^2-1{)}^2$. The two real branches approaching that point areThere is an important initial value problem qualification: the differential equation as printed has no classical solution through the specified initial point. Its right side is undefined at $y=0$, and these branches have an infinite one-sided derivative as $x\downarrow \sqrt{2}$. They satisfy the differential equation on $x>\sqrt{2}$ and extend continuously to the requested point. Multiplying by the vanishing denominator produces an implicit differential equation with a larger domain; it does not repair the original initial value problem.