Chapter 1.2, Problem 51E

To determine

To prove: The integral $x={\displaystyle {\int }_0^y\frac{1}{\sqrt{t^3+1}}}dt$ is an implicit solution of the initial-value problem $2\frac{d^{2}y}{d{x}^2}-3y^2=0$ where $y\left(0\right)=0$, $y^{\prime }\left(0\right)=1$ and $y\ge 0$.