Page 441 - Handbook Of Integral Equations

b

65. y(x)+ y(x + βt)f t, y(t) dt = A sin λx + B cos λx.
a
A solution:
y(x)= p sin λx + q cos λx, (1)

where p and q are roots of the following system of algebraic (or transcendental) equations:
b

p + p cos(λβt) – q sin(λβt) f t, p sin λt + q cos λt dt = A,
a (2)
b

q + q cos(λβt)+ p sin(λβt) f t, p sin λt + q cos λt dt = B.
a
Different solutions of system (2) generate different solutions (1) of the integral equation.

b

66. y(x)+ y(x + βt)f t, y(t) dt = g(x).
a
◦
1 .For g(x)= n A k exp(λ k x), the equation has a solution of the form
k=1
n

y(x)= B k exp(λ k x),
k=1
where the constants B k are determined from the nonlinear algebraic (or transcendental) system
%
B k + B k F k (B) – A k =0, k =1, ... , n,
b
n

%
%
B = {B 1 , ... , B n }, F k (B)= f t, B m exp(λ m t) exp(λ k βt) dt.
a
m=1
Different solutions of this system generate different solutions of the integral equation.
k
2 . For a polynomial right-hand side, g(x)= n A k x , the equation has a solution of the
◦
k=0
form
n
k
y(x)= B k x ,
k=0
where the constants B k can be found by the method of undetermined coefficients.
k
3 .For g(x)= e λx n A k x , the equation has a solution of the form
◦
k=0
n
k
y(x)= e λx B k x ,
k=0
where the constants B k can be found by the method of undetermined coefficients.

4 .For g(x)= n A k cos(λ k x), the equation has a solution of the form
◦
k=1
n n

y(x)= B k cos(λ k x)+ C k sin(λ k x),
k=1 k=1
where the constants B k and C k can be found by the method of undetermined coefficients.

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 421