Paper 2, Section II, 7C7 \mathrm{C}

Differential Equations | Part IA, 2017

Let y1y_{1} and y2y_{2} be two solutions of the differential equation

y(x)+p(x)y(x)+q(x)y(x)=0,<x<y^{\prime \prime}(x)+p(x) y^{\prime}(x)+q(x) y(x)=0, \quad-\infty<x<\infty

where pp and qq are given. Show, using the Wronskian, that

  • either there exist α\alpha and β\beta, not both zero, such that αy1(x)+βy2(x)\alpha y_{1}(x)+\beta y_{2}(x) vanishes for all xx,

  • or given x0,Ax_{0}, A and BB, there exist aa and bb such that y(x)=ay1(x)+by2(x)y(x)=a y_{1}(x)+b y_{2}(x) satisfies the conditions y(x0)=Ay\left(x_{0}\right)=A and y(x0)=By^{\prime}\left(x_{0}\right)=B.

Find power series y1y_{1} and y2y_{2} such that an arbitrary solution of the equation

y(x)=xy(x)y^{\prime \prime}(x)=x y(x)

can be written as a linear combination of y1y_{1} and y2y_{2}.

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