Page 209 - Handbook Of Integral Equations

7 .For f(x)= n A k sin(λ k x), a solution of the equation has the form
◦
k=0
n
A k

y(x)= 2 2 B ck sin(λ k x)+ B sk cos(λ k x) ,
B + B
k=0 ck sk
∞ ∞

B ck =1 + K(z) cos(λ k z) dz, B sk = K(z) sin(λ k z) dz.
0 0
8 .For f(x) = cos(λx) n A k exp(µ k x), a solution of the equation has the form
◦
k=0
n n

A k B ck A k B sk
y(x) = cos(λx) exp(µ k x) – sin(λx) exp(µ k x),
B 2 + B 2 B 2 + B 2
k=0 ck sk k=0 ck sk
∞ ∞

B ck =1 + K(z) exp(–µ k z) cos(λz) dz, B sk = K(z) exp(–µ k z) sin(λz) dz.
0 0
9 .For f(x) = sin(λx) n A k exp(µ k x), a solution of the equation has the form
◦
k=0
n n
A k B ck A k B sk

y(x) = sin(λx) exp(µ k x) + cos(λx) exp(µ k x),
B 2 + B 2 B 2 + B 2
k=0 ck sk k=0 ck sk
∞ ∞

B ck =1 + K(z) exp(–µ k z) cos(λz) dz, B sk = K(z) exp(–µ k z) sin(λz) dz.
0 0
∞

52. y(x)+ K(x – t)y(t) dt =0.
x
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 function K(–z) with
parameter –λ.
◦
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
⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
(1) r–1 (2) r–1
y (x)= x exp(α k x) cos(β k x), y (x)= x 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 188