Page 168 - Handbook Of Integral Equations

x

3. y(x)+ A cos[λ(x – t)]y(t) dt = f(x).
a
This is a special case of equation 2.9.34 with g(t)= A. Therefore, solving this integral
equation is reduced to solving the following second-order linear nonhomogeneous ordinary
differential equation with constant coefficients:

2
2

y + Ay + λ y = f xx + λ f, f = f(x),

x
xx
with the initial conditions

y(a)= f(a), y (a)= f (a) – Af(a).
x
x
1 . Solution with |A| >2|λ|:
◦
x

y(x)= f(x)+ R(x – t)f(t) dt,
a

A 2
1
2
R(x)=exp – Ax sinh(kx) – A cosh(kx) , k = 1 A – λ .
2

2 2k 4
◦
2 . Solution with |A| <2|λ|:
x
y(x)= f(x)+ R(x – t)f(t) dt,
a
A 2

1 2 1 2
R(x)=exp – Ax sin(kx) – A cos(kx) , k = λ – A .
2 4
2k
1
3 . Solution with λ = ± A:
◦
2
x
1 1 2
y(x)= f(x)+ R(x – t)f(t) dt, R(x)=exp – Ax A x – A .
2 2
a
x n

4. y(x)+ A k cos[λ k (x – t)] y(t) dt = f(x).
a
k=1
This integral equation is reduced to a linear nonhomogeneous ordinary differential equation
of order 2n with constant coefficients. Set
x
I k (x)= cos[λ k (x – t)]y(t) dt. (1)
a
Differentiating (1) with respect to x twice yields
x

I = y(x) – λ k sin[λ k (x – t)]y(t) dt,
k
a
x (2)

I = y (x) – λ 2 cos[λ k (x – t)]y(t) dt,

k x k
a
where the primes stand for differentiation with respect to x. Comparing (1) and (2), we see
that
2

I = y (x) – λ I k , I k = I k (x). (3)
k
x
k
© 1998 by CRC Press LLC
© 1998 by CRC Press LLC
Page 147