Page 270 - Handbook Of Integral Equations

4 . Solution with λ = λ 1,2 = λ ∗ and f 1 = f 2 = 0, where the characteristic value λ ∗ =
◦
4
is double:
2
2
(A + B)(b – a )
y(x)= f(x)+ Cy ∗ (x),
where C is an arbitrary constant and y ∗ (x) is an eigenfunction of the equation corresponding
to λ ∗ :
(A – B)(b + a)
y ∗ (x)= x – .
4A

b
4. y(x) – λ [A + B(x – t)]y(t) dt = f(x).
a
This is a special case of equation 4.9.8 with h(t)=1.
Solution:
y(x)= f(x)+ λ(A 1 + A 2 x),

where A 1 and A 2 are the constants determined by the formulas presented in 4.9.8.

b
5. y(x) – λ (Ax + Bt + C)y(t) dt = f(x).
a
This is a special case of equation 4.9.7 with g(x)= x and h(t)=1.
Solution:
y(x)= f(x)+ λ(A 1 x + A 2 ),
where A 1 and A 2 are the constants determined by the formulas presented in 4.9.7.

b
6. y(x)+ A |x – t| y(t) dt = f(x).
a
This is a special case of equation 4.9.36 with g(t)= A.
◦
1 . The function y = y(x) obeys the following second-order linear nonhomogeneous ordinary
differential equation with constant coefficients:

y +2Ay = f (x). (1)
xx
xx
The boundary conditions for (1) have the form (see 4.9.36)
y (a)+ y (b)= f (a)+ f (b),

x x x x
(2)

y(a)+ y(b)+(b – a)y (a)= f(a)+ f(b)+(b – a)f (a).
x x
Equation (1) under the boundary conditions (2) determines the solution of the original
integral equation.
◦
2 .For A < 0, the general solution of equation (1) is given by
x √

y(x)= C 1 cosh(kx)+ C 2 sinh(kx)+ f(x)+ k sinh[k(x – t)]f(t) dt, k = –2A, (3)
a
where C 1 and C 2 are arbitrary constants.
For A > 0, the general solution of equation (1) is given by
x
√
y(x)= C 1 cos(kx)+ C 2 sin(kx)+ f(x)– k sin[k(x – t)]f(t) dt, k = 2A. (4)
a
The constants C 1 and C 2 in solutions (3) and (4) are determined by conditions (2).

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 249