Page 436 - Handbook Of Integral Equations

k
2 .For g(x)=ln x n A k x , the equation has a solution of 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 can be found by the method of undetermined coefficients.

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

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

4 .For g(x)= n A k cos(λ k ln x), the equation has a solution of 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 can be found by the method of undetermined coefficients.

5 .For g(x)= n A k sin(λ k ln x), the equation has a solution of 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 can be found by the method of undetermined coefficients.

b

54. y(x)+ y(x – t)f t, y(t) dt =0.
a
◦
1 . A solution:
y(x)= ke Cx , (1)

where C is an arbitrary constant and the dependence k = k(C) is determined by the algebraic
(or transcendental) equation

b

Ct –Ct
1+ f t, ke e dt = 0. (2)
a
Each root of equation (2) generates a solution of the integral equation which has the form (1).

m
◦
2 . The equation has solutions of the form y(x)= n E m x , where the constants E m can
m=0
be found by the method of undetermined coefficients.

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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