= Optimal rectangle bound for a quartic separable initial-value problem
{title2=$y'=3x^2/(4y^3),\quad y(0)=b>0$}
For the <ordinary differential equation> $y'=3x^2/(4y^3)$ through $y(0)=b>0$, consider a symmetric existence rectangle $|x|\le h$, $|y-b|\le k<b$. Its maximum slope bound is $M=3h^2/[4(b-k)^3]$. The condition $Mh\le k$ becomes $h^3\le4k(b-k)^3/3$. Differentiating $k(b-k)^3$ shows its maximum occurs at $k=b/4$, giving
$$
h\le\frac{3^{2/3}b^{4/3}}4.
$$
This is the largest interval guaranteed by this particular centered-rectangle criterion. Direct <separation of variables> instead gives $y(x)=(b^4+x^3)^{1/4}$ on the maximal real interval $x>-b^{4/3}$. The comparison shows the difference between a sufficient local <Picard-Lindelöf theorem>[existence theorem] and the actual continuation interval. The endpoint obstruction is the zero of the denominator at $y=0$.
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