Page 244 - Handbook Of Integral Equations

where the primes denote the derivatives with respect to x. By comparing formulas (1) and (2),
we find the relation between I and I k :

k
2

I =2λ k y(x) – λ I k , I k = I k (x). (3)
k
k
◦
2 . With the aid of (1), the integral equation can be rewritten in the form
n

A k I k = f(x). (4)
k=1
Differentiating (4) with respect to x twice and taking into account (3), we find that
n n
2
σ 1 y(x) – A k λ I k = f (x), σ 1 =2 A k λ k . (5)

k xx
k=1 k=1
Eliminating the integral I n from (4) and (5) yields
n–1
2 2 2

σ 1 y(x)+ A k (λ – λ )I k = f (x)+ λ f(x). (6)
n k xx n
k=1
Differentiating (6) with respect to x twice and eliminating I n–1 from the resulting equation
with the aid of (6), we obtain a similar equation whose left-hand side is a second-order
n–2

linear differential operator (acting on y) with constant coefficients plus the sum B k I k .
k=1
If we successively eliminate I n–2 , I n–3 , ... , with the aid of double differentiation, then we
finally arrive at a linear nonhomogeneous ordinary differential equation of order 2(n–1) with
constant coefficients.
◦
3 . The right-hand side f(x) must satisfy certain conditions. To find these conditions, one
should set x = a in the integral equation and its derivatives. (Alternatively, these conditions
can be found by setting x = a and x = b in the integral equation and all its derivatives obtained
by means of double differentiation.)
b

k
13. sin x – sin t y(t) dt = f(x), 0 < k <1.
k
0
k
This is a special case of equation 3.8.3 with g(x) = sin x.
Solution:

1 d f (x)

x
y(x)= .
2k dx cos x sin k–1 x
The right-hand side f(x) must satisfy certain conditions. As follows from item 3 of equation
◦
3.8.3, the admissible general form of the right-hand side is given by

f(x)= F(x)+ Ax + B, A = –F (b), B = 1 2 bF (b) – F(0) – F(b) ,
x
x
where F(x) is an arbitrary bounded twice differentiable function (with bounded first deriva-
tive).
b
y(t)

14. dt = f(x), 0 < k <1.
a |sin(λx) – sin(λt)| k
This is a special case of equation 3.8.7 with g(x) = sin(λx)+β, where β is an arbitrary number.
a

15. k sin(λx) – t y(t) dt = f(x).

0
This is a special case of equation 3.8.5 with g(x)= k sin(λx).
a

16. x – k sin(λt) y(t) dt = f(x).

0
This is a special case of equation 3.8.6 with g(x)= k sin(λt).
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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