Page 288 - Handbook Of Integral Equations

1 . Solution with λ ≠ λ 1,2 :
◦
y(x)= f(x)+ λ(A 1 e βx + A 2 ),
where the constants A 1 and A 2 are given by

f 1 + λ f 1 ∆ β – (b – a)f 2 –f 2 + λ(f 2 ∆ β – f 1 ∆ 2β )
A 1 =
, A 2 =
,
λ (b – a)∆ 2β – ∆ 2 +1 λ (b – a)∆ 2β – ∆ 2 +1
2
2
β β
b b 1

βa
f 1 = f(x) dx, f 2 = f(x)e βx dx, ∆ β = (e βb – e ).
a a β
◦
2 . Solution with λ = λ 1 ≠ λ 2 and f 1 = f 2 =0:
y(x)= f(x)+ Cy 1 (x), y 1 (x)= e βx + 1 – λ 1 ∆ β ,
λ 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.
◦
b
βt
3. y(x) – λ (Ae βx + Be )y(t) dt = f(x).
a
The characteristic values of the equation:

2 2
(A + B)∆ β ± (A – B) ∆ +4AB(b – a)∆ 2β
β 1
βa
λ 1,2 =
, ∆ β = (e βb – e ).
2
2AB ∆ – (b – a)∆ 2β β
β
◦
1 . Solution with λ ≠ λ 1,2 :
y(x)= f(x)+ λ(A 1 e βx + A 2 ),
where the constants A 1 and A 2 are given by

Af 1 – ABλ f 1 ∆ β – (b – a)f 2
A 1 =
2 ,
2
ABλ ∆ – (b – a)∆ 2β – (A + B)λ∆ β +1
β
Bf 2 – ABλ(f 2 ∆ β – f 1 ∆ 2β )
A 2 =
,
2
2
ABλ ∆ – (b – a)∆ 2β – (A + B)λ∆ β +1
β
b b
f 1 = f(x) dx, f 2 = f(x)e βx dx.
a a
◦
2 . Solution with λ = λ 1 ≠ λ 2 and f 1 = f 2 =0:
y(x)= f(x)+ Cy 1 (x), y 1 (x)= e βx + 1 – Aλ 1 ∆ β ,
A(b – a)λ 1
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)∆ β
is double:
y(x)= f(x)+ Cy ∗ (x), y ∗ (x)= e βx – (A – B)∆ β ,
2A(b – a)
where C is an arbitrary constant and y ∗ (x) is an eigenfunction of the equation corresponding
to λ ∗ .
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 267