Page 443 - Handbook Of Integral Equations

b

67. y(x)+ y(ξ)f t, y(t) dt =0, ξ = xϕ(t).
a
1 . A solution:
◦
C
y(x)= kx , (1)
where C is an arbitrary constant and the dependence k = k(C) is determined by the algebraic
(or transcendental) equation

b

C C
1+ ϕ(t) f t, kt dt = 0. (2)
a
Each root of equation (2) generates a solution of the integral equation which has the form (1).
m
◦
2 . The equation has solutions of the form y(x)= n E m x , where the constants E m can
m=0
be found by the method of undetermined coefficients.
b

68. y(x)+ y(ξ)f t, y(t) dt = g(x), ξ = xϕ(t).
a
k
1 .For g(x)= n A k x , the equation has a solution of the form
◦
k=1
n
k
y(x)= B k x ,
k=1
where B k are roots of the algebraic (or transcendental) equations
%
B k + B k F k (B) – A k =0, k =1, ... , n,
n
b k
%
%

B = {B 1 , ... , B n }, F k (B)= ϕ(t) f t, B m t m dt.
a m=1
Different roots generate different solutions of the integral equation.
◦
◦
2 . For solutions with some other functions g(x), see items 2 –5 of equation 6.8.53.
◦
b

69. y(x)+ y(ξ)f t, y(t) dt =0, ξ = x + ϕ(t).
a
1 . A solution:
◦
y(x)= ke Cx , (1)
where C is an arbitrary constant and the dependence k = k(C) is determined by the algebraic
(or transcendental) equation
b

1+ e Cϕ(t) f t, ke Ct dt = 0. (2)
a
Each root of equation (2) generates a solution of the integral equation which has the form (1).

m
2 . The equation has a solution of the form y(x)= n E m x , where the constants E m can
◦
m=0
be found by the method of undetermined coefficients.

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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