Series Solutions · Hard problem

A polynomial coefficient of $y''$ that is never zero has no singular points anywhere in the complex plane, so there is no nearest singularity to limit the radius.

Because the equation has order $2$, every order-$2$ linear equation automatically has an infinite radius of convergence.

Because $x=0$ is an ordinary point, every ordinary point automatically gives an infinite radius regardless of the other coefficients.

Because $x^3$ vanishes at $x=0$, the vanishing of the coefficient there forces the radius to be infinite.