Mathematical Methods for Physics and Engineering : A Comprehensive Guide

SERIES SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS

(b) Find one series solution in powers ofz. Give a formal expression for a

second linearly independent solution.

(c) Deduce the values ofλfor which there is a polynomial solutionPN(z)of

degreeN. Evaluate the first four polynomials, normalised in such a way that

PN(0) = 1.

16.11 Find the general power series solution aboutz=0oftheequation

z

d^2 y

dz^2

+(2z−3)

dy

dz

+

4

z

y=0.

16.12 Find the radius of convergence of a series solution about the origin for the
equation (z^2 +az+b)y′′+2y= 0 in the following cases:

(a)a=5,b=6; (b)a=5,b=7.

Show that ifaandbare real and 4b>a^2 , then the radius of convergence is
always given byb^1 /^2.
16.13 For the equationy′′+z−^3 y= 0, show that the origin becomes a regular singular
point if the independent variable is changed fromztox=1/z. Hence find a
series solution of the formy 1 (z)=

∑∞

0 anz

−n.Bysettingy 2 (z)=u(z)y 1 (z)and

expanding the resulting expression fordu/dzin powers ofz−^1 , show thaty 2 (z)

has the asymptotic form

y 2 (z)=c

[

z+lnz−^12 +O

(

lnz

z

)]

,

wherecis an arbitrary constant.
16.14 Prove that the Laguerre equation,

z

d^2 y

dz^2

+(1−z)

dy

dz

+λy=0,

has polynomial solutionsLN(z)ifλis a non-negative integerN, and determine

the recurrence relationship for the polynomial coefficients. Hence show that an

expression forLN(z), normalised in such a way thatLN(0) =N!, is

LN(z)=

∑N

n=0

(−1)n(N!)^2

(N−n)!(n!)^2

zn.

EvaluateL 3 (z) explicitly.
16.15 The origin is an ordinary point of the Chebyshev equation,

(1−z^2 )y′′−zy′+m^2 y=0,

which therefore has series solutions of the formzσ

∑∞

0 anz

nforσ=0andσ=1.

(a) Find the recurrence relationships for theanin the two cases and show that

there exist polynomial solutionsTm(z):

(i) forσ=0,whenmis an even integer, the polynomial having^12 (m+2)

terms;

(ii) forσ=1,whenmis an odd integer, the polynomial having^12 (m+1)

terms.

(b)Tm(z) is normalised so as to haveTm(1) = 1. Find explicit forms forTm(z)

form=0, 1 , 2 ,3.