Page 342 - Handbook Of Integral Equations

∞
β
31. y(x) – K(xt)t y(t) dt = f(x).
0
The solution can be obtained with the aid of the inverse Mellin transform as follows:
1 c+i∞ –s
f(s)+ K(s)f(1 + β – s)
y(x)= x ds,
2πi 1 – K(s)K(1 + β – s)

c–i∞
where f and K stand for the Mellin transforms of the right-hand side and of the kernel of the

integral equation,

∞ ∞
f(s)= f(x)x s–1 dx, K(s)= K(x)x s–1 dx.

0 0

∞
λ µ
32. y(x) – g(xt)x t y(t) dt = f(x).
0
λ
This equation can be rewritten in the form of equation 4.9.31 by setting K(z)= z g(z) and
β = µ – λ.

∞ 1 x
33. y(x) – K y(t) dt =0.
0 t t
Eigenfunctions of this integral equation are determined by the roots of the following tran-
scendental (algebraic) equation for the parameter λ:

∞
1
K z λ–1 dz = 1. (1)
z
0
1 . For a real simple root λ n of equation (1), there is a corresponding eigenfunction
◦
y n (x)= x λ n .
2 . For a real root λ n of multiplicity r, there are corresponding r eigenfunctions
◦
y n1 (x)= x λ n , y n2 (x)= x λ n ln x, ... , y nr (x)= x λ n ln r–1 x.

3 . For a complex simple root λ n = α n + iβ n of equation (1), there is a corresponding pair
◦
of eigenfunctions
(1)
(2)
y (x)= x α n cos(β n ln x), y (x)= x α n sin(β n ln x).
n n
◦
4 . For a complex root λ n =α n +iβ n of multiplicity r, there are corresponding r eigenfunction
pairs
(2)
(1)
y (x)= x α n cos(β n ln x), y (x)= x α n sin(β n ln x),
n1 n1
(1)
(2)
y (x)= x α n ln x cos(β n ln x), y (x)= x α n ln x sin(β n ln x),
n2 n2
⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
(2)
(1)
y (x)= x α n ln r–1 x cos(β n ln x), y (x)= x α n ln r–1 x sin(β n ln x).
nr
nr
The general solution is the linear combination (with arbitrary constants) of the eigenfunc-
tions of the homogeneous integral equation.
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 321