Paper 2, Section II, B

Differential Equations | Part IA, 2007

(i) The function y(z)y(z) satisfies the equation

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

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

(ii) For the equation

4zy+2y+y=0,4 z y^{\prime \prime}+2 y^{\prime}+y=0,

classify the point z=0z=0 according to the definitions you gave in (i), and find the series solutions about z=0z=0. Identify these solutions in closed form.

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