Paper 2, Section II, A

Differential Equations | Part IA, 2013

The function y(x)y(x) satisfies the equation

y+p(x)y+q(x)y=0.y^{\prime \prime}+p(x) y^{\prime}+q(x) y=0 .

Give the definitions of the terms ordinary point, singular point, and regular singular point for this equation.

For the equation

xy+y=0x y^{\prime \prime}+y=0

classify the point x=0x=0 according to your definitions. Find the series solution about x=0x=0 which satisfies

y=0 and y=1 at x=0y=0 \quad \text { and } \quad y^{\prime}=1 \quad \text { at } x=0

For a second solution with y=1y=1 at x=0x=0, consider an expansion


where y0=1y_{0}=1 and xyn+1=ynx y_{n+1}^{\prime \prime}=-y_{n}. Find y1y_{1} and y2y_{2} which have yn(0)=0y_{n}(0)=0 and yn(1)=0y_{n}^{\prime}(1)=0. Comment on yy^{\prime} near x=0x=0 for this second solution.

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