Page 261 - Handbook Of Integral Equations

b
31. f(t)y(xt) dt = A sin(ln x).
a
Solution:
AI s AI c
y(x)= – cos(ln x)+ sin(ln x),
2
2
I + I 2 I + I 2
c s c s
b b
I c = f(t) cos(ln t) dt, I s = f(t) sin(ln t) dt.
a a
b

β
β
32. f(t)y(xt) dt = Ax cos(ln x)+ Bx sin(ln x).
a
Solution:
β
β
y(x)= px cos(ln x)+ qx sin(ln x),
where
AI c – BI s AI s + BI c
p = , q = ,
2
2
I + I 2 I + I 2
c s c s
b b
β
β
I c = f(t)t cos(ln t) dt, I s = f(t)t sin(ln t) dt.
a a
b

33. f(t)y(x – t) dt = Ax + B.
a
Solution:
y(x)= px + q,
where
A AI 1 B b b
p = , q = 2 + , I 0 = f(t) dt, I 1 = tf(t) dt.
I 0 I I 0 a a
0
b
34. f(t)y(x – t) dt = Ae λx .
a
Solution:
A λx b
y(x)= e , B = f(t) exp(–λt) dt.
B a
b
35. f(t)y(x – t) dt = A cos(λx).
a
Solution:
AI s AI c
y(x)= – sin(λx)+ cos(λx),
2
2
I + I s 2 I + I s 2
c
c
b b

I c = f(t) cos(λt) dt, I s = f(t) sin(λt) dt.
a a
b
36. f(t)y(x – t) dt = A sin(λx).
a
Solution:
AI c AI s
y(x)= sin(λx)+ cos(λx),
2
2
I + I 2 I + I 2
c s c s
b b
I c = f(t) cos(λt) dt, I s = f(t) sin(λt) dt.
a a
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 240