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As its documentation suggests in multiple places, rref is "mainly of academic interest" (read: "used only to explain Gaussian elimination to undergrads"), and is not a serious competitor of SVD-based algorithms in terms of stability. I recommend against it.


You can de-singularize Riccati equations $y'=y^2+a$ by setting $y=-\frac{u'}{u}$ to get $u''+au=0$. If $a$ is continuous on the integration interval, the solution will be well-behaved on that interval. Solving with $u(0)=1$ and $u'(0)=-y(0)=-2$ gives the plot with a root of $u$ and thus a singularity of $y$ at about $t=0.35230$. In any numerical integration,...

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