Page 262 - Handbook Of Integral Equations

b
37. f(t)y(x – t) dt = e µx (A sin λx + B cos λx).
a
Solution:
µx
y(x)= e (p sin λx + q cos λx),
where

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

I c = f(t)e –µt cos(λt) dt, I s = f(t)e –µt sin(λt) dt.
a a
b

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

y(x)= exp(λ k x), B k = f(t) exp(–λ k t) dt.
B k a
k=1
k
◦
2 . For a polynomial right-hand side, g(x)= n A k x , the solution has the form
k=0
n
k
y(x)= B k x ,
k=0
where the constants B k are found by the method of undetermined coefficients.
k
3 .For g(x)= e λx n A k x , the solution has the form
◦
k=0
n
k
y(x)= e λx B k x ,
k=0
where the constants B k are found by the method of undetermined coefficients.
◦
4 .For g(x)= n A k cos(λ k x), the solution 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 are found by the method of undetermined coefficients.

◦
5 .For g(x)= n A k sin(λ k x), the solution 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 are found by the method of undetermined coefficients.

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 241