Page 184 - Handbook Of Integral Equations

x

20. y(x)+ e µ(x–t) A 1 sin[λ 1 (x – t)] + A 2 sin[λ 2 (x – t)] y(t) dt = f(x).
a
The substitution w(x)= e –µx y(x) leads to an equation of the form 2.5.18:
x
–µx
w(x)+ A 1 sin[λ 1 (x – t)] + A 2 sin[λ 2 (x – t)] w(t) dt = e f(x).
a
x n

µ(x–t)
21. y(x)+ e A k sin[λ k (x – t)] y(t) dt = f(x).
a
k=1
The substitution w(x)= e –µx y(x) leads to an equation of the form 2.5.19:
n
x
w(x)+ A k sin[λ k (x – t)] w(t) dt = e –µx f(x).
a
k=1
x

22. y(x)+ A te µ(x–t) sin[λ(x – t)]y(t) dt = f(x).
a
Solution: x
Aλ µ(x–t)
y(x)= f(x)+ te u 1 (x)u 2 (t) – u 2 (x)u 1 (t) f(t) dt,
W
a
where u 1 (x), u 2 (x) is a fundamental system of solutions of the second-order linear ordinary
differential equation u + λ(Ax + λ)u = 0, and W is the Wronskian.
xx
Depending on the sign of Aλ, the functions u 1 (x) and u 2 (x) are expressed in terms of
Bessel functions or modified Bessel functions as follows:
if Aλ > 0, then
√ √
u 1 (x)= ξ 1/2 J 1/3 3 2 Aλ ξ 3/2 , u 2 (x)= ξ 1/2 Y 1/3 3 2 Aλ ξ 3/2 ,
W =3/π, ξ = x +(λ/A);

if Aλ < 0, then
√ √
u 1 (x)= ξ 1/2 I 1/3 3 2 –Aλ ξ 3/2 , u 2 (x)= ξ 1/2 K 1/3 3 2 –Aλ ξ 3/2 ,
3
W = – , ξ = x +(λ/A).
2
x
23. y(x)+ A xe µ(x–t) sin[λ(x – t)]y(t) dt = f(x).
a
Solution:
Aλ x µ(x–t)
y(x)= f(x)+ xe u 1 (x)u 2 (t) – u 2 (x)u 1 (t) f(t) dt,
W a
where u 1 (x), u 2 (x) is a fundamental system of solutions of the second-order linear ordinary
differential equation u + λ(Ax + λ)u = 0, and W is the Wronskian.
xx
The functions u 1 (x), u 2 (x), and W are specified in 2.7.22.

∞ √
24. y(x)+ A e µ(t–x) sin λ t – x y(t) dt = f(x).
x
√
–µx
This is a special case of equation 2.9.62 with K(x)= Ae sin λ –x .

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 163