Page 275 - Handbook Of Integral Equations

where the constants A 1 and A 2 are given by

1 1 1
f 1 – λ 4 1 ∆ 4 – f 2 ∆ 1 f 2 – λ 4 2 ∆ 4 – f 1 ∆ 7
f
f
A 1 = 1 1 1 , A 2 = 1 1 7 1 ,
2
2
λ 2 ∆ – ∆ 1 ∆ 7 – λ∆ 4 +1 λ 2 ∆ – ∆ 1 ∆ 7 – λ∆ 4 +1
16 4 7 2 16 4 7 2
b b
n
3
n
f 1 = f(x) dx, f 2 = x f(x) dx, ∆ n = b – a .
a a
◦
2 . Solution with λ = λ 1 ≠ λ 2 and f 1 = f 2 =0:

7
b – a 7
3
y(x)= f(x)+ Cy 1 (x), y 1 (x)= x + ,
7(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 . Solution with λ = λ 2 ≠ λ 1 and f 1 = f 2 =0:
◦

7
b – a 7
3
y(x)= f(x)+ Cy 2 (x), y 2 (x)= x – ,
7(b – a)
where C is an arbitrary constant and y 2 (x) is an eigenfunction of the equation corresponding
to the characteristic value λ 2 .
◦
4 . The equation has no multiple characteristic values.
b
3
3
21. y(x) – λ (x – t )y(t) dt = f(x).
a
The characteristic values of the equation:
1
.
λ 1,2 = ±
1
1 (a – b ) – (a – b )(b – a)
4
7
7
4 2
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 1 1
f
f 1 + λ 4 1 ∆ 4 – f 2 ∆ 1 –f 2 + λ 4 2 ∆ 4 – f 1 ∆ 7
f
A 1 = 1 1 , A 2 = 1 1 7 ,
λ 2 ∆ 1 ∆ 7 – ∆ 2 +1 λ 2 ∆ 1 ∆ 7 – ∆ 2 +1
7 16 4 7 16 4
b b
n
3
n
f 1 = f(x) dx, f 2 = x f(x) dx, ∆ n = b – a .
a a
2 . Solution with λ = λ 1 ≠ λ 2 and f 1 = f 2 =0:
◦
4
4
4 – λ 1 (b – a )
3
y(x)= f(x)+ Cy 1 (x), y 1 (x)= x + ,
4λ 1 (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.
◦
4 . The equation has no multiple characteristic values.
© 1998 by CRC Press LLC
© 1998 by CRC Press LLC
Page 254