Page 291 - Handbook Of Integral Equations

√
where a = –1 – 2λ, must be satisfied. In this case, the solution has the form
2
a +1 ∞
y(x)= f(x) – sin(at)f(x + t) dt, (–∞ < x < ∞).
2a 0
In the class of solutions not belonging to L 2 (–∞, ∞), the homogeneous equation (with
f(x) ≡ 0) has a nontrivial solution. In this case, the general solution of the corresponding
1
nonhomogeneous equation with λ ≤ – has the form
2
2
a +1 ∞
y(x)= C 1 sin(ax)+ C 2 cos(ax)+ f(x) – sin(a|x – t|)f(t) dt.
4a
–∞
•
Reference: F. D. Gakhov and Yu. I. Cherskii (1978).

b
15. y(x)+ A e λ|x–t| y(t) dt = f(x).
a
This is a special case of equation 4.9.37 with g(t)= A.
1 . The function y = y(x) obeys the following second-order linear nonhomogeneous ordinary
◦
differential equation with constant coefficients:
2

y xx + λ(2A – λ)y = f (x) – λ f(x). (1)
xx
The boundary conditions for (1) have the form (see 4.9.37)
y (a)+ λy(a)= f (a)+ λf(a),

x
x
(2)

y (b) – λy(b)= f (b) – λf(b).
x
x
Equation (1) under the boundary conditions (2) determines the solution of the original
integral equation.
2 .For λ(2A – λ) < 0, the general solution of equation (1) is given by
◦
2Aλ x
y(x)= C 1 cosh(kx)+ C 2 sinh(kx)+ f(x) – sinh[k(x – t)] f(t) dt,
k a (3)

k = λ(λ – 2A),
where C 1 and C 2 are arbitrary constants.
For λ(2A – λ) > 0, the general solution of equation (1) is given by
x
2Aλ
y(x)= C 1 cos(kx)+ C 2 sin(kx)+ f(x) – sin[k(x – t)] f(t) dt,
k a (4)

k = λ(2A – λ).
For λ =2A, the general solution of equation (1) is given by

x
y(x)= C 1 + C 2 x + f(x) – 4A 2 (x – t)f(t) dt. (5)
a
The constants C 1 and C 2 in solutions (3)–(5) are determined by conditions (2).
3 . In the special case a = 0 and λ(2A – λ) > 0, the solution of the integral equation is given
◦
by formula (4) with
A(kI c – λI s ) λ A(kI c – λI s )
C 1 = , C 2 = – ,
(λ – A) sin µ – k cos µ k (λ – A) sin µ – k cos µ
b b

k = λ(2A – λ), µ = bk, I s = sin[k(b – t)]f(t) dt, I c = cos[k(b – t)]f(t) dt.
0 0

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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