Page 276 - Handbook Of Integral Equations

b
3
3
22. y(x) – λ (Ax + Bt )y(t) dt = f(x).
a
The characteristic values of the equation:

1 1 2 2 4
4
4 (A + B)∆ 4 ± 16 (A – B) ∆ + AB∆ 1 ∆ 7 n n
7
λ 1,2 = , ∆ n = b – a .
1
2
2AB 1 ∆ – ∆ 1 ∆ 7
16 4 7
1 . Solution with λ ≠ λ 1,2 :
◦
3
y(x)= f(x)+ λ(A 1 x + A 2 ),
where the constants A 1 and A 2 are given by
1
Af 1 – ABλ f
A 1 = 1 1 4 1 ∆ 4 – f 2 ∆ 1 ,

1
2
ABλ 2 ∆ – ∆ 1 ∆ 7 – λ(A + B)∆ 4 +1
16 4 7 4
1 1
Bf 2 – ABλ f
7
A 2 = 1 1 4 2 ∆ 4 – f 1 ∆ 7 ,

1
2
ABλ 2 ∆ – ∆ 1 ∆ 7 – λ(A + B)∆ 4 +1
16 4 7 4
b b
3
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:
◦
4
4
4 – λ 1 A(b – a )
3
y(x)= f(x)+ Cy 1 (x), y 1 (x)= x + ,
4λ 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.
8
4 . Solution with λ = λ 1,2 = λ ∗ and f 1 = f 2 = 0, where λ ∗ = is the double
◦
4
4
(A + B)(b – a )
characteristic value:
4
4
(A – B)(b – a )
3
y(x)= f(x)+ Cy ∗ (x), y ∗ (x)= x – ,
8A(b – a)
where C is an arbitrary constant and y ∗ (x) is an eigenfunction of the equation corresponding
to λ ∗ .
b
2
3
23. y(x) – λ (xt – t )y(t) dt = f(x).
a
2
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.
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 255