= Solution
For the suggested <substitution>, put $y=xv(x)$. Then the <product rule> gives $y'=v+xv'$, while the right side of the <ordinary differential equation> becomes $v+v^{-3}$. Cancelling $v$ leaves the <separable differential equation>
$$
x\frac{dv}{dx}=v^{-3},\qquad v^3\,dv=\frac{dx}{x}.
$$
Integration 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:
$$
\boxed{y(x)=x(1+4\log x)^{1/4},\qquad x>e^{-1/4}.}
$$
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.
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