Page 183 - Handbook Of Integral Equations

x

14. y(x) – e µ(x–t) A cos(kx)+ B – AB(x – t) cos(kx) y(t) dt = f(x).
a
Solution:
x
y(x)= f(x)+ e µ(x–t) M(x, t)f(t) dt,
a
G(x) B 2 x B(x–s) A
M(x, t)=[A cos(kx)+ B] + e G(s) ds, G(x)=exp sin(kx) .
G(t) G(t) t k
x

15. y(x)+ e µ(x–t) A cos(kt)+ B + AB(x – t) cos(kt) y(t) dt = f(x).
a
Solution:
x

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

x
16. y(x) – A e µt sin(λx)y(t) dt = f(x).
a
µt
This is a special case of equation 2.9.2 with g(x)= A sin(λx) and h(t)= e .
x
17. y(x) – A e µx sin(λt)y(t) dt = f(x).
a
This is a special case of equation 2.9.2 with g(x)= Ae µx and h(t) = sin(λt).

x
18. y(x)+ A e µ(x–t) sin[λ(x – t)]y(t) dt = f(x).
a
1 . Solution with λ(A + λ)>0:
◦
x

Aλ µ(x–t)
y(x)= f(x) – e sin[k(x – t)]f(t) dt, where k = λ(A + λ).
k
a
2 . Solution with λ(A + λ)<0:
◦
Aλ x µ(x–t)
y(x)= f(x) – e sinh[k(x – t)]f(t) dt, where k = –λ(λ + A).
k a
3 . Solution with A = –λ:
◦
x

y(x)= f(x)+ λ 2 (x – t)e µ(x–t) f(t) dt.
a
x
3
19. y(x)+ A e µ(x–t) sin [λ(x – t)]y(t) dt = f(x).
a
The substitution w(x)= e –µx y(x) leads to an equation of the form 2.5.17:
x
3
w(x)+ A sin [λ(x – t)]w(t) dt = e –µx f(x).
a

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 162