Page 279 - Handbook Of Integral Equations

b

n
n
33. y(x) – λ (Ax + Bt )y(t) dt = f(x), n =1, 2, ...
a
The characteristic values of the equation:

2 2
(A + B)∆ n ± (A – B) ∆ +4AB∆ 0 ∆ 2n 1 n+1 n+1
n
λ 1,2 = , ∆ n = (b – a ).
2
2AB(∆ – ∆ 0 ∆ 2n ) n +1
n
1 . Solution with λ ≠ λ 1,2 :
◦
n
y(x)= f(x)+ λ(A 1 x + A 2 ),
where the constants A 1 and A 2 are given by
Af 1 – ABλ(f 1 ∆ n – f 2 ∆ 0 )
A 1 = ,
2
2
ABλ (∆ – ∆ 0 ∆ 2n ) – (A + B)λ∆ n +1
n
Bf 2 – ABλ(f 2 ∆ n – f 1 ∆ 2n )
A 2 = ,
2
2
ABλ (∆ – ∆ 0 ∆ 2n ) – (A + B)λ∆ n +1
n
b b
n
f 1 = f(x) dx, f 2 = x f(x) dx.
a a
◦
2 . Solution with λ = λ 1 ≠ λ 2 and f 1 = f 2 =0:
n
y(x)= f(x)+ Cy 1 (x), y 1 (x)= x + 1 – Aλ 1 ∆ n ,
Aλ 1 ∆ 0
where C is an arbitrary constant and y 1 (x) is an eigenfunction of the equation corresponding
to the characteristic value λ 1 .
◦
◦
3 . The solution with λ = λ 2 ≠ λ 1 and f 1 = f 2 = 0 is given by the formulas of item 2 in
which one must replace λ 1 and y 1 (x)by λ 2 and y 2 (x), respectively.
2
4 . Solution with λ=λ 1,2 =λ ∗ and f 1 =f 2 =0, where the characteristic value λ ∗ =
◦
(A + B)∆ n
is double:
n
y(x)= f(x)+ Cy ∗ (x), y ∗ (x)= x – (A – B)∆ n .
2A∆ 0
Here C is an arbitrary constant and y ∗ (x) is an eigenfunction of the equation corresponding
to λ ∗ .
b

m
34. y(x) – λ (x – t)t y(t) dt = f(x), m =1, 2, ...
a
m
This is a special case of equation 4.9.8 with A =0, B = 1, and h(t)= t .
Solution:
y(x)= f(x)+ λ(A 1 + A 2 x),
where A 1 and A 2 are the constants determined by the formulas presented in 4.9.8.

b
m
35. y(x) – λ (x – t)x y(t) dt = f(x), m =1, 2, ...
a
m
This is a special case of equation 4.9.10 with A =0, B = 1, and h(x)= x .
Solution:
m
y(x)= f(x)+ λ(A 1 x m+1 + A 2 x ),
where A 1 and A 2 are the constants determined by the formulas presented in 4.9.10.

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 258