 Solution

For the suggested substitution, put $y=xv(x)$. Then the product rule gives $y^{\prime }=v+x{v}^{\prime }$, while the right side of the ordinary differential equation becomes $v+v^{-3}$. Cancelling $v$ leaves the separable differential equationIntegration yields $v^4/4=\log x+C$. The initial condition gives $v(1)=1$, so $v^4=1+4\log x$. The positive root is forced by the initial value and continuity:This is the maximal connected interval containing $1$ on which the real solution exists. At the left endpoint $y=0$, where the original ordinary differential equation is undefined; extending the fourth-root formula further left would also cease to be real.