The function $y(x)$ satisfies the equation

$$y^{\prime \prime}+p(x) y^{\prime}+q(x) y=0 .$$

Give the definitions of the terms ordinary point, singular point, and regular singular point for this equation.

For the equation

$$x y^{\prime \prime}+y=0$$

classify the point $x=0$ according to your definitions. Find the series solution about $x=0$ which satisfies

$$y=0 \quad \text { and } \quad y^{\prime}=1 \quad \text { at } x=0$$

For a second solution with $y=1$ at $x=0$, consider an expansion

$$y(x)=y_{0}(x)+y_{1}(x)+y_{2}(x)+\ldots,$$

where $y_{0}=1$ and $x y_{n+1}^{\prime \prime}=-y_{n}$. Find $y_{1}$ and $y_{2}$ which have $y_{n}(0)=0$ and $y_{n}^{\prime}(1)=0$. Comment on $y^{\prime}$ near $x=0$ for this second solution.