Paper 2, Section II, A

Differential Equations | Part IA, 2016

The function y(x)y(x) satisfies

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

What does it mean to say that the point x=0x=0 is (i) an ordinary point and (ii) a regular singular point of this differential equation? Explain what is meant by the indicial equation at a regular singular point. What can be said about the nature of the solutions in the neighbourhood of a regular singular point in the different cases that arise according to the values of the roots of the indicial equation?

State the nature of the point x=0x=0 of the equation

xy+(xm+1)y(m1)y=0x y^{\prime \prime}+(x-m+1) y^{\prime}-(m-1) y=0

Set y(x)=xσn=0anxny(x)=x^{\sigma} \sum_{n=0}^{\infty} a_{n} x^{n}, where a00a_{0} \neq 0, and find the roots of the indicial equation.

(a) Show that one solution of ()(*) with m0,1,2,m \neq 0,-1,-2, \cdots is

y(x)=xm(1+n=1(1)nxn(m+n)(m+n1)(m+1))y(x)=x^{m}\left(1+\sum_{n=1}^{\infty} \frac{(-1)^{n} x^{n}}{(m+n)(m+n-1) \cdots(m+1)}\right)

and find a linearly independent solution in the case when mm is not an integer.

(b) If mm is a positive integer, show that ()(*) has a polynomial solution.

(c) What is the form of the general solution of ()(*) in the case m=0m=0 ? [You do not need to find the general solution explicitly.]

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