Differential equation with variable coefficients

Question

I saw this differential equation somewhere $$y''+xy=0$$ it was solved using the substitution $$y=x^\alpha u$$ where $\alpha$, is a constant. My question is how can we substitute for $y$ with an expression containing the independent variable $x$? Is it because we put $x$ raised to power $\alpha$ so in that case we haven't made any assumptions about $y$ depending on $x$, as $\alpha$ may be any value depending on the solution itself?