Paper 2, Section II, B

Differential Equations | Part IA, 2014

(a) Let y1(x)y_{1}(x) be a solution of the equation

d2ydx2+p(x)dydx+q(x)y=0\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 y2(x)=y_{2}(x)= v(x)y1(x)v(x) y_{1}(x), derive an ordinary differential equation for v(x)v(x).

(b) Consider the equation

(1x2)d2ydx22xdydx+2y=0,1<x<1.\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)y(x) satisfying the conditions

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

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