Optimal rectangle bound for a quartic separable initial-value problem ($y^{\prime }=3x^2/(4y^3),y(0)=b>0$)

For the ordinary differential equation $y^{\prime }=3x^2/(4y^3)$ 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 $h^3\le 4k(b-k{)}^3/3$. Differentiating $k(b-k{)}^3$ shows its maximum occurs at $k=b/4$, givingThis 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 existence theorem and the actual continuation interval. The endpoint obstruction is the zero of the denominator at $y=0$.