Mathematical Methods for Physics and Engineering : A Comprehensive Guide

EIGENFUNCTION METHODS FOR DIFFERENTIAL EQUATIONS

interval, if necessary by changing the signs of all eigenvalues. Fora≤x 1 ≤x 2 ≤b,

establish the identity

(λn−λm)

∫x 2

x 1

ρynymdx=

[

ynpy′m−ympy′n

]x 2

x 1.

Deduce that ifλn>λmthenyn(x) must change sign between two successive zeros
ofym(x).
[ The reader may find it helpful to illustrate this result by sketching the first few
eigenfunctions of the systemy′′+λy=0,withy(0) =y(π) = 0, and the Legendre
polynomialsPn(z)forn=2, 3 , 4 , 5 .]
17.4 Show that the equation

y′′+aδ(x)y+λy=0,

withy(±π)=0andareal, has a set of eigenvaluesλsatisfying

tan(π

√

λ)=

2

√

λ

a

.

Investigate the conditions under which negative eigenvalues,λ=−μ^2 ,withμ
real, are possible.
17.5 Use the properties of Legendre polynomials to carry out the following exercises.

(a) Find the solution of (1−x^2 )y′′− 2 xy′+by=f(x), valid in the range

− 1 ≤x≤1 and finite atx= 0, in terms of Legendre polynomials.

(b) Ifb=14andf(x)=5x^3 , find the explicit solution and verify it by direct

substitution.

[ The first six Legendre polynomials are listed in Subsection 18.1.1. ]
17.6 Starting from the linearly independent functions 1,x,x^2 ,x^3 ,..., in the range
0 ≤x<∞, find the first three orthogonal functionsφ 0 ,φ 1 andφ 2 ,withrespect
to the weight functionρ(x)=e−x. By comparing your answers with the Laguerre
polynomials generated by the recurrence relation (18.115), deduce the form of
φ 3 (x).
17.7 Consider the set of functions,{f(x)}, of the real variablex, defined in the interval
−∞<x<∞,that→0atleastasquicklyasx−^1 asx→±∞. For unit weight
function, determine whether each of the following linear operators is Hermitian
when acting upon{f(x)}:

(a)

d

dx

+x;(b)−i

d

dx

+x^2 ;(c)ix

d

dx

;(d)i

d^3

dx^3

.

17.8 A particle moves in a parabolic potential in which its natural angular frequency
of oscillation is^12 .Attimet= 0 it passes through the origin with velocityv.It
is then suddenly subjected to an additional acceleration, of +1 for 0≤t≤π/2,
followed by−1forπ/ 2 <t≤π. At the end of this period it is again at the
origin. Apply the results of the worked example in section 17.5 to show that

v=−

8

π

∑∞

m=0

1

(4m+2)^2 −^14

≈− 0. 81.

17.9 Find an eigenfunction expansion for the solution, with boundary conditions
y(0) =y(π) = 0, of the inhomogeneous equation

d^2 y

dx^2

+κy=f(x),