(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

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