Paper 2, Section II, A

Differential Equations | Part IA, 2012

Find the first three non-zero terms in the series solutions y1(x)y_{1}(x) and y2(x)y_{2}(x) for the differential equation

x2y2xy+(2x2)y=0x^{2} y^{\prime \prime}-2 x y^{\prime}+\left(2-x^{2}\right) y=0

that satisfy

y1(0)=a and y1(0)=0y2(0)=0 and y2(0)=2b\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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