Power substitution reducing a differential equation to constant coefficients ($y=x^{n}z$)

On an interval where the substitution is real and $x\ne 0$, a factor formed by exponentiation can remove inverse powers from the coefficients of a linear ordinary differential equation. Forthe substitution $y=x^{n}z$ givesIt becomes a constant-coefficient differential equation exactly when $2n+b=0$, $an+d=0$ and $n(n-1)+bn+e=0$. Derive this by the product rule; checking only the coefficient of $z^{\prime }$ is insufficient.