Page 287 - Handbook Of Integral Equations

4.2. Equations Whose Kernels Contain Exponential
Functions

4.2-1. Kernels Containing Exponential Functions

b
βt
1. y(x) – λ (e βx + e )y(t) dt = f(x).
a
The characteristic values of the equation:
β
λ 1,2 = .
e βb – e βa ± 1 β(b – a)(e 2βb – e 2βa )
2
◦
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 =
,
2
2
λ ∆ – (b – a)∆ 2β – 2λ∆ β +1 λ ∆ – (b – a)∆ 2β – 2λ∆ β +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:

e 2βb – e 2βa
βx
y(x)= f(x)+ Cy 1 (x), y 1 (x)= e + ,
2β(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:

e 2βb – e 2βa
βx
y(x)= f(x)+ Cy 2 (x), y 2 (x)= e – ,
2β(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
βt
2. y(x) – λ (e βx – e )y(t) dt = f(x).
a
The characteristic values of the equation:

β
λ 1,2 = ± .
1
βa 2
(e βb – e ) – β(b – a)(e 2βb – e 2βa )
2

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 266