Paper 2, Section II, A

Differential Equations | Part IA, 2021

For a linear, second order differential equation define the terms ordinary point, singular point and regular singular point.

For a,bRa, b \in \mathbb{R} and bZb \notin \mathbb{Z} consider the following differential equation

xd2y dx2+(bx)dy dxay=0.x \frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}+(b-x) \frac{\mathrm{d} y}{\mathrm{~d} x}-a y=0 .

Find coefficients cm(a,b)c_{m}(a, b) such that the function y1=M(x,a,b)y_{1}=M(x, a, b), where

M(x,a,b)=m=0cm(a,b)xmM(x, a, b)=\sum_{m=0}^{\infty} c_{m}(a, b) x^{m}

satisfies ()(*). By making the substitution y=x1bu(x)y=x^{1-b} u(x), or otherwise, find a second linearly independent solution of the form y2=x1bM(x,α,β)y_{2}=x^{1-b} M(x, \alpha, \beta) for suitable α,β\alpha, \beta.

Suppose now that b=1b=1. By considering a limit of the form

limb1y2y1b1\lim _{b \rightarrow 1} \frac{y_{2}-y_{1}}{b-1}

or otherwise, obtain two linearly independent solutions to ()(*) in terms of MM and derivatives thereof.

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