Page 350 - Handbook Of Integral Equations

b

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

where the coefficients p and q are given by

AI c + BI s BI c – AI s
p = , q = ,
2
I + I 2 I + I 2
2
c s c s

b b
I c =1 + f(t) cos(λβt) dt, I s = f(t) sin(λβt) dt.
a a
b
50. y(x)+ f(t)y(x + βt) dt = g(x).
a
n
1 .For g(x)= A k exp(λ k x), a solution of the equation has the form
◦
k=1
n b
A k
y(x)= exp(λ k x), B k =1 + f(t) exp(βλ k t) dt.
B k a
k=1
n k
◦
2 . For polynomial right-hand side of the equation, g(x)= A k x , a solution has the form
k=0
n

k
y(x)= B k x ,
k=0
where the constants B k can be found by the method of undetermined coefficients.
n
k
◦
3 .For g(x)= e λx A k x , a solution of the equation has the form
k=0
n
k
λx
y(x)= e B k x ,
k=0
where the constants B k can be found by the method of undetermined coefficients.
n
4 .For g(x)= A k cos(λ k x), a solution of the equation has 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.
n
◦
5 .For g(x)= A k sin(λ k x), a solution of the equation has 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 329