= Tangent comparison for finite-time Riccati blow-up
{title2=$y'=1+x^2+y^2,\quad y(0)=1\Longrightarrow x_*\leq\pi/4$}
Compare the displayed <ordinary differential equation> with $u'=1+u^2$, $u(0)=1$, whose solution is $u=\tan(x+\pi/4)$. Their difference satisfies $w'=(y+u)w+x^2$, $w(0)=0$. The <integrating factor> formula makes $w\geq0$ on the common forward existence interval. Since $u$ diverges at $\pi/4$, the increasing solution $y$ has a maximal forward endpoint $0<x_*\leq\pi/4$. 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 $|x|\leq1$, $|y|\leq10$, a bound $|1+x^2+y^2|\leq102$ and distance nine to the nearer horizontal box edge give the explicit guaranteed interval $|x|\leq3/34$.
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