Page 170 - Handbook Of Integral Equations

x

m
k
10. y(x)+ A x cos (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) = cos (λt).
x

11. y(x) – A cos(kx)+ B – AB(x – t) cos(kx) y(t) dt = f(x).
a
This is a special case of equation 2.9.7 with λ = B and g(x)= A cos(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 cos(kx)+ B] + e G(s) ds, G(x)=exp sin(kx) .
G(t) G(t) t k
x

12. y(x)+ A cos(kt)+ B + AB(x – t) cos(kt) y(t) dt = f(x).
a
This is a special case of equation 2.9.8 with λ = B and g(t)= A cos(kt).
Solution:

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

∞
√

13. y(x)+ A cos λ t – x y(t) dt = f(x).
x
√
This is a special case of equation 2.9.62 with K(x)= A cos λ –x .

2.5-2. Kernels Containing Sine

x
14. y(x) – A sin(λx)y(t) dt = f(x).
a
This is a special case of equation 2.9.2 with g(x)= A sin(λx) and h(t)=1.
Solution:
x

A
y(x)= f(x)+ A sin(λx)exp cos(λt) – cos(λx) f(t) dt.
a λ
x

15. y(x) – A sin(λt)y(t) dt = f(x).
a
This is a special case of equation 2.9.2 with g(x)= A and h(t) = sin(λt).
Solution:
x
A
y(x)= f(x)+ A sin(λt)exp cos(λt) – cos(λx) f(t) dt.
a λ

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 149