Mathematical Methods for Physics and Engineering : A Comprehensive Guide

17.7 EXERCISES

whereκis a constant and

f(x)=

{

x 0 ≤x≤π/ 2 ,

π−xπ/ 2 <x≤π.

17.10 Consider the following two approaches to constructing a Green’s function.

(a) Find those eigenfunctionsyn(x) of the self-adjoint linear differential operator

d^2 /dx^2 that satisfy the boundary conditionsyn(0) =yn(π) = 0, and hence

construct its Green’s functionG(x, z).

(b) Construct the same Green’s function using a method based on the comple-

mentary function of the appropriate differential equation and the boundary

conditions to be satisfied at the position of theδ-function, showing that it is

G(x, z)=

{

x(z−π)/π 0 ≤x≤z,

z(x−π)/π z≤x≤π.

(c) By expanding the function given in (b) in terms of the eigenfunctionsyn(x),

verify that it is the same function as that derived in (a).

17.11 The differential operatorLis defined by

Ly=−

d

dx

(

ex

dy

dx

)

−^14 exy.

Determine the eigenvaluesλnof the problem

Lyn=λnexyn 0 <x< 1 ,

with boundary conditions

y(0) = 0,

dy

dx

+^12 y=0 atx=1.

(a) Find the corresponding unnormalisedyn, and also a weight functionρ(x)with

respect to which theynare orthogonal. Hence, select a suitable normalisation

for theyn.

(b) By making an eigenfunction expansion, solve the equation

Ly=−ex/^2 , 0 <x< 1 ,

subject to the same boundary conditions as previously.

17.12 Show that the linear operator

L≡^14 (1 +x^2 )^2

d^2

dx^2

+^12 x(1 +x^2 )

d

dx

+a,

acting upon functions defined in− 1 ≤x≤1 and vanishing at the end-points of

the interval, is Hermitian with respect to the weight function (1 +x^2 )−^1.

By making the change of variablex=tan(θ/2), find two even eigenfunctions,

f 1 (x)andf 2 (x), of the differential equation

Lu=λu.

17.13 By substitutingx=expt, find the normalised eigenfunctionsyn(x)andthe
eigenvaluesλnof the operatorLdefined by

Ly=x^2 y′′+2xy′+^14 y, 1 ≤x≤e,

withy(1) =y(e) = 0. Find, as a series

∑

anyn(x), the solution ofLy=x−^1 /^2.