Page 123 - Handbook Of Integral Equations

k
2 .For f(x)= e λx n A k x , the solution has the form
◦
k=0
n
k
y(x)= e λx B k x ,
k=0
where the constants B k are found by the method of undetermined coefficients. The solution
can also be obtained by the formula given in 1.9.29 (item 3 ).
◦
3 .For f(x)= n A k exp(λ k x), the solution has the form
◦
k=0
n
A k
∞
y(x)= exp(λ k x), B k = K(–z) exp(λ k z) dz.
B k 0
k=0
k
4 .For f(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
5 .For f(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.

6 .For f(x)= n A k cos(λ k x), the solution has the form
◦
k=0
n
A k

y(x)= 2 2 B ck cos(λ k x)+ B sk sin(λ k x) ,
B + B
k=0 ck sk
∞ ∞

B ck = K(–z) cos(λ k z) dz, B sk = K(–z) sin(λ k z) dz.
0 0
◦
7 .For f(x)= n A k sin(λ k x), the solution has the form
k=0
n
A k
y(x)= B ck sin(λ k x) – B sk cos(λ k x) ,
B 2 + B 2
k=0 ck sk
∞ ∞

B ck = K(–z) cos(λ k z) dz, B sk = K(–z) sin(λ k z) dz.
0 0
◦
8 . For arbitrary right-hand side f = f(x), the solution of the integral equation can be
calculated by the formula
1 c+i∞ ˜ px
f(p)
y(x)= e dp,
˜
2πi c–i∞ k(–p)
∞ ∞

˜
˜
f(p)= f(x)e –px dx, k(–p)= K(–z)e pz dz.
0 0
˜
˜
To calculate f(p) and k(–p), it is convenient to use tables of Laplace transforms, and to
determine y(x), tables of inverse Laplace transforms.
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 101