Page 333 - Handbook Of Integral Equations

b
13. y(x) – λ [g(x)g(t) – h(x)h(t)]y(t) dt = f(x).
a
The characteristic values of the equation:
2 2
2
2
s 1 – s 3 + (s 1 + s 3 ) – 4s s 1 – s 3 – (s 1 + s 3 ) – 4s
λ 1 = 2 , λ 2 = 2 ,
2
2
2(s – s 1 s 3 ) 2(s – s 1 s 3 )
2 2
where
b b b

2
2
s 1 = g (x) dx, s 2 = g(x)h(x) dx, s 3 = h (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
f 1 + λ(f 1 s 3 – f 2 s 2 ) –f 2 + λ(f 2 s 1 – f 1 s 2 )
A 1 = , A 2 = ,
2
2
2
2
(s – s 1 s 3 )λ – (s 1 – s 3 )λ +1 (s – s 1 s 3 )λ – (s 1 – s 3 )λ +1
2 2
b b

f 1 = f(x)g(x) dx, f 2 = f(x)h(x) dx.
a a
◦
2 . Solution with λ = λ 1 ≠ λ 2 and f 1 = f 2 =0:
1 – λ 1 s 1
y(x)= f(x)+ Cy 1 (x), y 1 (x)= g(x)+ h(x),
λ 1 s 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 ∗ =
◦
s 1 – s 3
is double:
s 1 + s 3
y(x)= f(x)+ Cy ∗ (x), y ∗ (x)= g(x) – h(x),
2s 2
where C is an arbitrary constant and y ∗ (x) is an eigenfunction of the equation corresponding
to λ ∗ .
b

14. y(x) – λ [Ag(x)g(t)+ Bh(x)h(t)]y(t) dt = f(x).
a
The characteristic values of the equation:
2
2
As 1 + Bs 3 ± (As 1 – Bs 3 ) +4ABs
λ 1,2 = 2 2 ,
2AB(s 1 s 3 – s )
2
where
b b b

2
2
s 1 = g (x) dx, s 2 = g(x)h(x) dx, s 3 = h (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)],
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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