Page 174 - Handbook Of Integral Equations

x

m
k
22. y(x) – A sin (λx) sin (µt)y(t) dt = f(x).
a
m
k
This is a special case of equation 2.9.2 with g(x)= A sin (λx) and h(t) = sin (µt).
x

23. y(x)+ A t sin[λ(x – t)]y(t) dt = f(x).
a
This is a special case of equation 2.9.36 with g(t)= At.
Solution:
Aλ x
y(x)= f(x)+ t 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

24. y(x)+ A x sin[λ(x – t)]y(t) dt = f(x).
a
This is a special case of equation 2.9.37 with g(x)= Ax and h(t)=1.
Solution:
Aλ x
y(x)= f(x)+ x 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 2.5.23.
x

m
k
25. y(x)+ A t sin (λx)y(t) dt = f(x).
a
m
k
This is a special case of equation 2.9.2 with g(x)= –A sin (λx) and h(t)= t .
x

k
m
26. y(x)+ A x sin (λt)y(t) dt = f(x).
a
k
m
This is a special case of equation 2.9.2 with g(x)= –Ax and h(t) = sin (λt).
x

27. y(x) – A sin(kx)+ B – AB(x – t) sin(kx) y(t) dt = f(x).
a
This is a special case of equation 2.9.7 with λ = B and g(x)= A sin(kx).
Solution:
x

y(x)= f(x)+ R(x, t)f(t) dt,
a
G(x) B 2 x B(x–s) A
R(x, t)=[A sin(kx)+ B] + e G(s) ds, G(x)=exp – cos(kx) .
G(t) G(t) t k
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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