Paper 2, Section II, A

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.

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