Page 265 - Handbook Of Integral Equations

b
45. f(t)y(ξ) dt = g(x), ξ = xϕ(t).
a
k
1 .For g(x)= n A k x , the solution of the equation has the form
◦
k=0
n b
A k k
k
y(x)= x , B k = f(t) ϕ(t) dt.
B k a
k=0
2 .For g(x)= n A k x , the solution has the form
λ k
◦
k=0
n b
A k λ k

y(x)= x , B k = f(t) ϕ(t) dt.
λ k
B k a
k=0
k
◦
3 .For g(x)=ln x n A k x , the solution has the form
k=0
n n
k k
y(x)=ln x B k x + C k x ,
k=0 k=0

where the constants B k and C k are found by the method of undetermined coefficients.
k
◦
n
4 .For g(x)= A k ln x) , the solution has the form
k=0
n

k
y(x)= B k ln x) ,
k=0
where the constants B k are found by the method of undetermined coefficients.

◦
5 .For g(x)= n A k cos(λ k ln x), the solution has the form
k=1
n n

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

6 .For g(x)= n A k sin(λ k ln x), the solution has the form
◦
k=1
n n

y(x)= B k cos(λ k ln x)+ C k sin(λ k ln 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 244