Page 140 - Handbook Of Integral Equations

◦
3 . For special λ = λ n ,(n =1, 2, ... ), the solution differs in one term and has the form
n–1 N
A m m A m m λ n n n
¯
y(x)= x + x – A n x ln x + Cx ,
1 – (λ n /λ m ) 1 – (λ n /λ m ) λ n
m=0 m=n+1
1 z ln zdz –1
n

¯
where λ n = √ .
0 a + bz 2
◦
4 . For arbitrary f(x), expandable into power series, the formulas of item 2 can be used, in
◦
which one should set N = ∞. In this case, the radius of convergence of the solution y(x)is
equal to the radius of convergence of f(x).
x y(t) dt
48. y(x)+ λ = f(x).
a (x – t) 3/4
This equation admits solution by quadratures (see equation 2.1.60 and Section 9.4-2).
2.1-7. Kernels Containing Arbitrary Powers

x
λ
49. y(x)+ A (x – t)t y(t) dt = f(x).
a
λ
This is a special case of equation 2.9.4 with g(t)= At .
Solution:
x

A λ
y(x)= f(x)+ y 1 (x)y 2 (t) – y 2 (x)y 1 (t) t f(t) dt,
W
a
where y 1 (x), y 2 (x) is a fundamental system of solutions of the second-order linear homo-
λ
geneous ordinary differential equation y + Ax y = 0; the functions y 1 (x) and y 2 (x) are
xx
expressed in terms of Bessel functions or modified Bessel functions, depending on the sign
of A:
For A >0,
√ √
2q √ A q √ A q λ +2
W = , y 1 (x)= xJ 1 x , y 2 (x)= xY 1 x , q = ,
π 2q q 2q q 2
For A <0,
√ √
√ |A| q √ |A| q λ +2
W = –q, y 1 (x)= xI 1 x , y 2 (x)= xK 1 x , q = .
2q q 2q q 2
x

λ µ
50. y(x)+ A x t y(t) dt = f(x).
a
λ
µ
This is a special case of equation 2.9.2 with g(x)= –Ax and h(t)= t (λ and µ are arbitrary
numbers).
Solution:
x
y(x)= f(x) – R(x, t)f(t) dt,
a

A
λ µ λ+µ+1 λ+µ+1
Ax t exp for λ + µ +1 ≠ 0,
 t – x
R(x, t)= λ + µ +1
 λ–A µ+A
Ax t for λ + µ + 1=0.
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 119