Series solutions of linear second-order ODEs

Use power series methods to solve $y'' + 2 x^3 y = 0$ at the point $x_0 = 0$.

Power-series methods also work for nonhomogeneous equations.

1. Use power series methods to solve $y'' - x y = \frac{1}{1-x}$ at the point $x_0 = 0$. Hint: Recall the geometric series.
2. Now solve for the initial condition $y(0)=0$, $y'(0)=0$.

Attempt to solve $x^2 y'' - y = 0$ at $x_0 = 0$ using the power series method of this section ($x_0$ is a singular point). Can you find at least one solution? Can you find more than one solution?