Page 325 - Handbook Of Integral Equations

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

2 2
(A + B)g 1 ± (A – B) g +4AB(b – a)g 2
1
λ 1,2 = ,
2
2AB[g – (b – a)g 2 ]
1
where

b b
2
g 1 = g(x) dx, g 2 = g (x) dx.
a a
1 . Solution with λ ≠ λ 1,2 :
◦
y(x)= f(x)+ λ[A 1 g(x)+ A 2 ],
where the constants A 1 and A 2 are given by
Af 1 –λAB[f 1 g 1 –(b–a)f 2 ] Bf 2 –λAB(f 2 g 1 –f 1 g 2 )
A 1 = 2 , A 2 = 2 ,
2
2
AB[g –(b–a)g 2 ]λ –(A+B)g 1 λ+1 AB[g –(b–a)g 2 ]λ –(A+B)g 1 λ+1
1
1
b b
f 1 = f(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 Ag 1
y(x)= f(x)+ Cy 1 (x), y 1 (x)= g(x)+ ,
λ 1 A(b – a)
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)g 1
is double:
(A – B)g 1
y(x)= f(x)+ Cy ∗ (x), y ∗ (x)= g(x) – .
2A(b – a)
Here C is an arbitrary constant and y ∗ (x) is an eigenfunction of the equation corresponding
to λ ∗ .

b

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

2
s 1 + s 3 ± (s 1 – s 3 ) +4(b – a)s 2
λ 1,2 = ,
2[s 1 s 3 – (b – a)s 2 ]
where

b b b
s 1 = g(x) dx, s 2 = g(x)h(x) dx, s 3 = h(x) dx.
a a a

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 304