Paper 2, Section II, B

(a) Let $y_{1}(x)$ be a solution of the equation

$\frac{d^{2} y}{d x^{2}}+p(x) \frac{d y}{d x}+q(x) y=0$

Assuming that the second linearly independent solution takes the form $y_{2}(x)=$ $v(x) y_{1}(x)$, derive an ordinary differential equation for $v(x)$.

(b) Consider the equation

$\left(1-x^{2}\right) \frac{d^{2} y}{d x^{2}}-2 x \frac{d y}{d x}+2 y=0, \quad-1<x<1 .$

By inspection or otherwise, find an explicit solution of this equation. Use the result in (a) to find the solution $y(x)$ satisfying the conditions

$y(0)=\frac{d y}{d x}(0)=1$

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