Page 353 - Handbook Of Integral Equations

b

54. y(x)+ f(t)y(xt) dt = A ln x + B.
a
A solution:
y(x)= p ln x + q,

where
b b
A B AI l
p = , q = – 2 , I 0 = f(t) dt, I l = f(t)ln tdt.
1+ I 0 1+ I 0 (1 + I 0 ) a a

b
β
55. y(x)+ f(t)y(xt) dt = Ax ln x.
a
A solution:
β β
y(x)= px ln x + qx ,
where
b b
A AI 2 β β
p = , q = – 2 , I 1 = f(t)t dt, I 2 = f(t)t ln tdt.
1+ I 1 (1 + I 1 )
a a
b
56. y(x)+ f(t)y(xt) dt = A cos(ln x).
a
A solution:

AI c AI s
y(x)= cos(ln x)+ sin(ln x),
2
2
I + I s 2 I + I 2 s
c
c
b b
I c =1 + f(t) cos(ln t) dt, I s = f(t) sin(ln t) dt.
a a
b
57. y(x)+ f(t)y(xt) dt = A sin(ln x).
a
A solution:
AI s AI c
y(x)= – cos(ln x)+ sin(ln x),
I + I 2 s I + I s 2
2
2
c
c

b b
I c =1 + f(t) cos(ln t) dt, I s = f(t) sin(ln t) dt.
a a
b

β
β
58. y(x)+ f(t)y(xt) dt = Ax cos(λ ln x)+ Bx sin(λ ln x).
a
A solution:
β
β
y(x)= px cos(λ ln x)+ qx sin(λ 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 =1 + f(t)t cos(λ ln t) dt, I s = f(t)t sin(λ ln t) dt.
a a
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 332