 Solution

Multiplying the ordinary differential equation by two suggests using the dependent variable $Y=y^2$, whose derivative is $Y^{\prime }=2y{y}^{\prime }$. On the interval through $x=1$, this gives the linear ordinary differential equationAfter integration, $Y=x^2+Cx$. The initial condition $Y(1)=4$ gives $C=3$. As $y(1)>0$, continuity selects the positive branch:The interval $x>0$ is the maximal real interval through the given initial condition: at $x=0$ the formula approaches $0$, but its derivative is unbounded, so it does not extend through zero as a differentiable solution of a differential equation.