Page 169 - Handbook Of Integral Equations

With the aid of (1), the integral equation can be rewritten in the form

n

y(x)+ A k I k = f(x). (4)
k=1
Differentiating (4) with respect to x twice taking into account (3) yields

n n
2
y (x)+ σ n y (x) – A k λ I k = f (x), σ n = A k . (5)

k
x
xx
xx
k=1 k=1
Eliminating the integral I n from (4) and (5), we obtain
n–1
2 2 2
2

y (x)+ σ n y (x)+ λ y(x)+ A k (λ – λ )I k = f (x)+ λ f(x). (6)

n
xx
n
k
xx
x
n
k=1
Differentiating (6) with respect to x twice followed by eliminating I n–1 from the resulting
expression with the aid of (6) yields a similar equation whose left-hand side is a fourth-
n–2
order differential operator (acting on y) with constant coefficients plus the sum B k I k .
k=1
Successively eliminating the terms I n–2 , I n–3 , ... using double differentiation and formula (3),
we finally arrive at a linear nonhomogeneous ordinary differential equation of order 2n with
constant coefficients.
The initial conditions for y(x) can be obtained by setting x = a in the integral equation
and all its derivative equations.
x
cos(λx)
5. y(x) – A y(t) dt = f(x).
a cos(λt)
Solution:
x
cos(λx)
y(x)= f(x)+ A e A(x–t) f(t) dt.
a cos(λt)
x cos(λt)
6. y(x) – A y(t) dt = f(x).
a cos(λx)
Solution:
x cos(λt)
y(x)= f(x)+ A e A(x–t) f(t) dt.
a cos(λx)
x

m
k
7. y(x) – A cos (λx) cos (µt)y(t) dt = f(x).
a
k
m
This is a special case of equation 2.9.2 with g(x)= A cos (λx) and h(t) = cos (µt).
x
8. y(x)+ A t cos[λ(x – t)]y(t) dt = f(x).
a
This is a special case of equation 2.9.34 with g(t)= At.
x

m
k
9. y(x)+ A t cos (λx)y(t) dt = f(x).
a
k
m
This is a special case of equation 2.9.2 with g(x)= –A cos (λx) and h(t)= t .
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 148