Page 266 - Handbook Of Integral Equations

b
46. f(t)y(ξ) dt = g(x), ξ = x + ϕ(t).
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 ϕ(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
λx
y(x)= e 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.
k
6 .For g(x) = cos(λx) n A k x , the solution has the form
◦
k=0
n n
k k
y(x) = cos(λx) B k x + sin(λx) C k x ,
k=0 k=0
where the constants B k and C k are found by the method of undetermined coefficients.
k
7 .For g(x) = sin(λx) n A k x , the solution has the form
◦
k=0
n n
k k
y(x) = cos(λx) B k x + sin(λx) C k x ,
k=0 k=0
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 245