Page 297 - Handbook Of Integral Equations

b
m
k
23. y(x) – λ x sinh (βt)y(t) dt = f(x).
a
m
k
This is a special case of equation 4.9.1 with g(x)= x and h(t) = sinh (βt).

b
24. y(x) – λ [A + B(x – t) sinh(βt)]y(t) dt = f(x).
a
This is a special case of equation 4.9.8 with h(t) = sinh(βt).
b
25. y(x) – λ [A + B(x – t) sinh(βx)]y(t) dt = f(x).
a
This is a special case of equation 4.9.10 with h(x) = sinh(βx).

b
26. y(x)+ A sinh(λ|x – t|)y(t) dt = f(x).
a
This is a special case of equation 4.9.38 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 4.9.38)
sinh[λ(b – a)]ϕ (b) – λ cosh[λ(b – a)]ϕ(b)= λϕ(a),

x
ϕ(x)= y(x) – f(x). (2)
sinh[λ(b – a)]ϕ (a)+ λ cosh[λ(b – a)]ϕ(a)= –λϕ(b),

x
Equation (1) under the boundary conditions (2) determines the solution of the original
integral equation.
2
2 .For λ(2A – λ)= –k < 0, the general solution of equation (1) is given by
◦
x
2Aλ
y(x)= C 1 cosh(kx)+ C 2 sinh(kx)+ f(x) – sinh[k(x – t)]f(t) dt, (3)
k a
where C 1 and C 2 are arbitrary constants.
2
For λ(2A – λ)= k > 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. (4)
k
a
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).

b
27. y(x)+ A t sinh(λ|x – t|)y(t) dt = f(x).
a
This is a special case of equation 4.9.38 with g(t)= At. The solution of the integral equation
can be written via the Bessel functions (or modified Bessel functions) of order 1/3.

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 276