Page 216 - Handbook Of Integral Equations

k
2 .For g(x)=ln x n A k x , a 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
3 .For g(x)= A k ln x) , a solution of the equation has the form
◦
k=0
n

k
y(x)= B k ln x) ,
k=0
where the B k are found by the method of undetermined coefficients.
4 .For g(x)= n A k cos(λ k ln x), a solution of the equation 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 B k and C k are found by the method of undetermined coefficients.

5 .For g(x)= n A k sin(λ k ln x) a solution of the equation 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 B k and C k are found by the method of undetermined coefficients.

6 . For arbitrary right-hand side g(x), the transformation
◦
z
z
–τ
–z
x = e , t = e , y(x)= e w(z), f(ξ)= F(ln ξ), g(x)= e G(z)
leads to an equation with difference kernel of the form 2.9.62:
∞

w(z)+ F(z – τ)w(τ) dτ = G(z).
z
◦
7 . For arbitrary right-hand side g(x), the solution of the integral equation can be expressed
via the inverse Mellin transform (see Section 7.3-1).

2.10. Some Formulas and Transformations

Let the solution of the integral equation
x

y(x)+ K(x, t)y(t) dt = f(x) (1)
a
have the form
x
y(x)= f(x)+ R(x, t)f(t) dt. (2)
a

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 195