Page 336 - Handbook Of Integral Equations

b
17. y(x) – λ [Ag(x)h(t)+ Bh(x)g(t)]y(t) dt = f(x).
a
The characteristic values of the equation:

2 2
(A + B)s 1 ± (A – B) s +4ABs 2 s 3
1
λ 1,2 = ,
2
2AB(s – s 2 s 3 )
1
where

b b b
2
2
s 1 = h(x)g(x) dx, s 2 = h (x) dx, s 3 = g (x) dx.
a a a
◦
1 . Solution with λ ≠ λ 1,2 :
y(x)= f(x)+ λ[A 1 g(x)+ A 2 h(x)],
where the constants A 1 and A 2 are given by
Af 1 – λAB(f 1 s 1 – f 2 s 2 ) Bf 2 – λAB(f 2 s 1 – f 1 s 3 )
A 1 = , A 2 = ,
2
2
2
2
AB(s – s 2 s 3 )λ – (A + B)s 1 λ +1 AB(s – s 2 s 3 )λ – (A + B)s 1 λ +1
1 1

b b
f 1 = f(x)h(x) dx, f 2 = f(x)g(x) dx.
a a
◦
2 . Solution with λ = λ 1 ≠ λ 2 and f 1 = f 2 =0:
1 – λ 1 As 1
y(x)= f(x)+ Cy 1 (x), y 1 (x)= g(x)+ h(x),
λ 1 As 2
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)s 1
is double:
(A – B)s 1
y(x)= f(x)+ Cy ∗ (x), y ∗ (x)= g(x) – h(x).
2As 2
Here C is an arbitrary constant and y ∗ (x) is an eigenfunction of the equation corresponding
to λ ∗ .
b
18. y(x) – λ [g 1 (x)h 1 (t)+ g 2 (x)h 2 (t)]y(t) dt = f(x).
a
The characteristic values of the equation λ 1 and λ 2 are given by

2
s 11 + s 22 ± (s 11 – s 22 ) +4s 12 s 21
λ 1,2 = ,
2(s 11 s 22 – s 12 s 21 )
provided that the integrals

b b b b
s 11 = h 1 (x)g 1 (x) dx, s 12 = h 1 (x)g 2 (x) dx, s 21 = h 2 (x)g 1 (x) dx, s 22 = h 2 (x)g 2 (x) dx
a a a a
are convergent.

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 315