Tangent comparison for finite-time Riccati blow-up ($y^{\prime }=1+x^2+y^2,y(0)=1⟹x_{\ast }\le \pi /4$)

Compare the displayed ordinary differential equation with $u^{\prime }=1+u^2$, $u(0)=1$, whose solution is $u=\tan (x+\pi /4)$. Their difference satisfies $w^{\prime }=(y+u)w+x^2$, $w(0)=0$. The integrating factor formula makes $w\ge 0$ on the common forward existence interval. Since $u$ diverges at $\pi /4$, the increasing solution $y$ has a maximal forward endpoint $0<x_{\ast }\le \pi /4$. If $y$ remained bounded at this finite endpoint, local existence with the smooth right side would continue it, a contradiction. Hence $y\to +\infty$. The Picard-Lindelöf theorem guarantees local existence, not existence through such a finite-time blow-up. On $|x|\le 1$, $|y|\le 10$, a bound $|1+x^2+y^2|\le 102$ and distance nine to the nearer horizontal box edge give the explicit guaranteed interval $|x|\le 3/34$.