Page 352 - Handbook Of Integral Equations

b
51. y(x)+ f(t)y(xt) dt =0.
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)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
(1) (2)
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
(2)
(1)
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
⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
(1)
(2)
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.
For equations 4.9.52–4.9.58, only particular solutions are given. To obtain the general solution,
one must add the particular solution to the general solution of the corresponding homogeneous
equation 4.9.51.

b
52. y(x)+ f(t)y(xt) dt = Ax + B.
a
A solution:
b b
A B
y(x)= x + , I 0 = f(t) dt, I 1 = tf(t) dt.
1+ I 1 1+ I 0 a a

b
β
53. y(x)+ f(t)y(xt) dt = Ax .
a
A solution:
b
A β β
y(x)= x , B =1 + f(t)t dt.
B a
* In the equations below that contain y(xt) in the integrand, the arguments can have, for example, the domain (a) 0 ≤ x ≤ 1,
0 ≤ t ≤ 1 for a = 0 and b = 1, (b) 1 ≤ x < ∞,1 ≤ t < ∞ for a = 1 and b = ∞, (c) 0 ≤ x < ∞,0 ≤ t < ∞ for a = 0 and b = ∞,
or (d) a ≤ t ≤ b,0 ≤ x < ∞ for a and b such that 0 ≤ a < b ≤ ∞. Case (d) is a special case of (c) if f(t) is nonzero only on
the interval a ≤ t ≤ b.

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 331