Paper 2, Section II, A

Find the first three non-zero terms in the series solutions $y_{1}(x)$ and $y_{2}(x)$ for the differential equation

$x^{2} y^{\prime \prime}-2 x y^{\prime}+\left(2-x^{2}\right) y=0$

that satisfy

$\begin{array}{llc} y_{1}^{\prime}(0)=a & \text { and } & y_{1}^{\prime \prime}(0)=0 \\ y_{2}^{\prime}(0)=0 & \text { and } & y_{2}^{\prime \prime}(0)=2 b \end{array}$

Identify these solutions in closed form.

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