Page 146 - Handbook Of Integral Equations

In this case, the original integral equation has the solution

x

µ 1 (x–t) µ 2 (x–t)
y(x)= f(x)+ E 1 e + E 2 e f(t) dt,
a
where
µ 1 µ 1 – λ µ 2 µ 2 – λ
E 1 = A + B , E 2 = A + B .
µ 2 – µ 1 µ 2 – µ 1 µ 1 – µ 2 µ 1 – µ 2
◦
3 .If D < 0, then equation (2) has the complex conjugate roots
√
1
µ 1 = σ + iβ, µ 2 = σ – iβ, σ = (λ – A – B), β = 1 –D.
2 2
In this case, the original integral equation has the solution

x
y(x)= f(x)+ E 1 e σ(x–t) cos[β(x – t)] + E 2 e σ(x–t) sin[β(x – t)] f(t) dt,
a
where
1
E 1 = –A – B, E 2 = (–Aσ – Bσ + Bλ).
β
x

λt
5. y(x)+ A (e λx – e )y(t) dt = f(x).
a
λx
This is a special case of equation 2.9.5 with g(x)= Ae .
Solution:
1 x

y(x)= f(x)+ u (x)u (t) – u (x)u (t) f(t) dt,
1
2
1
2
W
a
where the primes denote differentiation with respect to the argument specified in the paren-
theses, and u 1 (x), u 2 (x) is a fundamental system of solutions of the second-order linear
λx
homogeneous ordinary differential equation u +Aλe u = 0; the functions u 1 (x) and u 2 (x)
xx
are expressed in terms of Bessel functions or modified Bessel functions, depending on the
sign of A:
For Aλ >0,
√ √
λ 2 Aλ λx/2 2 Aλ λx/2
W = , u 1 (x)= J 0 e , u 2 (x)= Y 0 e ,
π λ λ
For Aλ <0,
√ √
λ 2 |Aλ| λx/2 2 |Aλ| λx/2
W = – , u 1 (x)= I 0 e , u 2 (x)= K 0 e .
2 λ λ
x

λx λt
6. y(x)+ Ae + Be y(t) dt = f(x).
a
For B = –A, see equation 2.2.5. This is a special case of equation 2.9.6 with g(x)= Ae λx and
λt
h(t)= Be .
x
Differentiating the original integral equation followed by substituting Y (x)= y(t) dt
a
yields the second-order linear ordinary differential equation
λx
λx

Y xx +(A + B)e Y + Aλe Y = f (x) (1)
x
x
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 125