Page 355 - Handbook Of Integral Equations

n
k
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) , (3)
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 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), (4)
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 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), (5)
k=1 k=1
where the constants B k and C k can be found by the method of undetermined coefficients.
Remark. A linear combination of eigenfunctions of the corresponding homogeneous
equation (see 4.9.59) can be added to solutions (1)–(5).

4.10. Some Formulas and Transformations

Let the solution of the integral equation

b

y(x)+ K(x, t)y(t) dt = f(x) (1)
a
have the form
b
y(x)= f(x)+ R(x, t)f(t) dt. (2)
a
Then the solution of the more complicated integral equation

b g(x)

y(x)+ K(x, t) y(t) dt = f(x) (3)
a g(t)
has the form

b g(x)
y(x)= f(x)+ R(x, t) f(t) dt. (4)
a g(t)
Below are formulas for the solutions of integral equations of the form (3) for some specific func-
tions g(x). In all cases, it is assumed that the solution of equation (1) is known and is given
by (2).

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 334