Chapter 6.1, Problem 5E

In Problems $1-6$, determine the largest interval $\left(a,b\right)$ for which Theorem $1$ guarantees the existence of a unique solution on $\left(a,b\right)$ to the given initial value problem.

$x\sqrt{x+1}{y}^{‴}-y^{\prime }+xy=0$;

$y\left(1/2\right)=y^{\prime }\left(1/2\right)=-1$, $y^{″}\left(1/2\right)=1$