Page 205 - Handbook Of Integral Equations

2 . Let w = w(x) be the solution of the simpler auxiliary equation with a = 0 and f ≡ 1:
◦
x

w(x)+ K(x – t)w(t) dt = 1. (2)
0
Then the solution of the original integral equation with arbitrary f = f(x) is expressed via the
solution of the auxiliary equation (2) as

x x
d

y(x)= w(x – t)f(t) dt = f(a)w(x – a)+ w(x – t)f (t) dt.
t
dx
a a
•
Reference: R. Bellman and K. L. Cooke (1963).
x
41. y(x)+ K(x – t)y(t) dt =0.
–∞
Eigenfunctions of this integral equation are determined by the roots of the following tran-
scendental (algebraic) equation for the parameter λ:

∞

K(z)e –λz dz = –1. (1)
0
The left-hand side of this equation is the Laplace transform of the kernel of the integral
equation.
1 . For a real simple root λ k of equation (1) there is a corresponding eigenfunction
◦

y k (x)=exp(λ k x).

2 . For a real root λ k of multiplicity r there are corresponding r eigenfunctions
◦
y k1 (x) = exp(λ k x), y k2 (x)= x exp(λ k x), ... , y kr (x)= x r–1 exp(λ k x).

3 . For a complex simple root λ k = α k + iβ k of equation (1) there is a corresponding
◦
eigenfunction pair
(1)
(2)
y (x) = exp(α k x) cos(β k x), y (x)=exp(α k x) sin(β k x).
k k
◦
4 . For a complex root λ k = α k +iβ k of multiplicity r there are corresponding r eigenfunction
pairs
(1) (2)
y (x)=exp(α k x) cos(β k x), y (x) = exp(α k x) sin(β k x),
k1 k1
(1) (2)
y (x)= x exp(α k x) cos(β k x), y (x)= x exp(α k x) sin(β k x),
k2 k2
⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
(2)
(1)
y (x)= x r–1 exp(α k x) cos(β k x), y (x)= x r–1 exp(α k x) sin(β k x).
kr kr
The general solution is the combination (with arbitrary constants) of the eigenfunctions
of the homogeneous integral equation.

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 184