Page 354 - Handbook Of Integral Equations

b
59. y(x)+ f(t)y(ξ) dt =0, ξ = xϕ(t).
a
Eigenfunctions of this integral equation are determined by the roots of the following tran-
scendental (or algebraic) equation for λ:
b
λ
f(t)[ϕ(t)] dt = –1. (1)
a
1 . For a real (simple) root λ k of equation (1), there is a corresponding eigenfunction
◦
y k (x)= x .
λ k
2 . For a real root λ k of multiplicity r, there are corresponding r eigenfunctions
◦
y k1 (x)= x , y k2 (x)= x λ k ln x, ... , y kr (x)= x λ k ln r–1 x.
λ k
3 . For a complex (simple) root λ k = α k + iβ k of equation (1), there is a corresponding pair
◦
of eigenfunctions
(2)
(1)
y (x)= x α k cos(β k ln x), y (x)= x α k sin(β k ln x).
k k
4 . For a complex root λ k = α k + iβ k of multiplicity r, there are corresponding r pairs of
◦
eigenfunctions
(1) (2)
y (x)= x α k cos(β k ln x), y (x)= x α k sin(β k ln x),
k1 k1
(1)
(2)
y (x)= x α k ln x cos(β k ln x), y (x)= x α k ln x sin(β k ln x),
k2 k2
⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
(2)
(1)
y (x)= x α k ln r–1 x cos(β k ln x), y (x)= x α k ln r–1 x sin(β k ln x).
kr kr
The general solution is the linear combination (with arbitrary constants) of the eigenfunc-
tions of the homogeneous integral equation.
b

β
60. y(x)+ f(t)y(ξ) dt = Ax , ξ = xϕ(t).
a
A solution:
b
A β β
y(x)= x , B =1 + f(t)[ϕ(t)] dt.
B
a
It is assumed that B ≠ 0. A linear combination of eigenfunctions of the corresponding
homogeneous equation (see 4.9.59) can be added to this solution.
b

61. y(x)+ f(t)y(ξ) dt = g(x), ξ = xϕ(t).
a
n
k
◦
1 .For g(x)= A k x , a solution of the equation has the form
k=0
n b

A k k k
y(x)= x , B k =1 + f(t)[ϕ(t)] dt. (1)
B k a
k=0
n
k
2 .For g(x)=ln x A k x , a solution has the form
◦
k=0
n n

k
k
y(x)=ln x B k x + C k x , (2)
k=0 k=0
where the constants B k and C k can be found by the method of undetermined coefficients.
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 333