Page 214 - Handbook Of Integral Equations

◦
2 . For a real root λ k of multiplicity r there are corresponding r eigenfunctions
y k1 (x)= x , y k2 (x)= x λ k ln x, ... , y kr (x)= x λ k ln r–1 x.
λ k
◦
3 . For a complex simple root λ k = α k + iβ k of equation (1) there is a corresponding
eigenfunction pair
(1)
(2)
y (x)= x α k cos(β k ln x), y (x)= x α k sin(β k ln x).
k k
4 . For a complex root λ k = α k +iβ k of multiplicity r there are corresponding r eigenfunction
◦
pairs
(1) (2)
y (x)= x α k cos(β k ln x), y (x)= x α k sin(β k ln x),
k1 k1
(1) (2)
y (x)= x α k ln x cos(β k ln x), y (x)= x α k ln x sin(β k ln x),
k2 k2
⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
(2)
(1)
y (x)= x α k ln r–1 x cos(β k ln x), y (x)= x α k ln r–1 x sin(β k ln x).
kr kr
The general solution is the combination (with arbitrary constants) of the eigenfunctions
of the homogeneous integral equation.
For equations 2.9.64–2.9.71, only particular solutions are given. To obtain the general solu-
tion, one must add the general solution of the corresponding homogeneous equation 2.9.63 to the
particular solution.
x 1 t
64. y(x)+ f y(t) dt = Ax + B.
0 x x
A solution:
A B 1 1
y(x)= x + , I 0 = f(t) dt, I 1 = tf(t) dt.
1+ I 1 1+ I 0 0 0
x 1 t
β
65. y(x)+ f y(t) dt = Ax .
0 x x
A solution:
A β 1 β
y(x)= x , B =1 + f(t)t dt.
B 0
x 1 t
66. y(x)+ f y(t) dt = A ln x + B.
0 x x
A solution:
y(x)= p ln x + q,
where
A B AI l 1 1
p = , q = – 2 , I 0 = f(t) dt, I l = f(t)ln tdt.
1+ I 0 1+ I 0 (1 + I 0 ) 0 0
x
1 t β

67. y(x)+ f y(t) dt = Ax ln x.
0 x x
A solution:
β
β
y(x)= px ln x + qx ,
where
A AI 2 1 β 1 β
p = , q = – 2 , I 1 = f(t)t dt, I 2 = f(t)t ln tdt.
1+ I 1 (1 + I 1 )
0 0

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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