Paper 2, Section II, $7 \mathrm{C}$

Let $y_{1}$ and $y_{2}$ be two solutions of the differential equation

$y^{\prime \prime}(x)+p(x) y^{\prime}(x)+q(x) y(x)=0, \quad-\infty<x<\infty$

where $p$ and $q$ are given. Show, using the Wronskian, that

either there exist $\alpha$ and $\beta$, not both zero, such that $\alpha y_{1}(x)+\beta y_{2}(x)$ vanishes for all $x$,

or given $x_{0}, A$ and $B$, there exist $a$ and $b$ such that $y(x)=a y_{1}(x)+b y_{2}(x)$ satisfies the conditions $y\left(x_{0}\right)=A$ and $y^{\prime}\left(x_{0}\right)=B$.

Find power series $y_{1}$ and $y_{2}$ such that an arbitrary solution of the equation

$y^{\prime \prime}(x)=x y(x)$

can be written as a linear combination of $y_{1}$ and $y_{2}$.

*Typos? Please submit corrections to this page on GitHub.*