Page 346 - Handbook Of Integral Equations

Eliminating the integral terms from (1) and (2), we arrive at a second-order ordinary
differential equation for y = y(x),
2
2
y +2λg(x)y + λ y = f (x)+ λ f(x). (3)

xx xx
2 . Let us derive the boundary conditions for equation (3). We assume that the limits of
◦
integration satisfy the conditions –∞ < a < b < ∞. By setting x = a and x = b in (1), we
obtain two consequences
b
y(a)+ sin[λ(t – a)]g(t)y(t) dt = f(a),
a
b (4)
y(b)+ sin[λ(b – t)]g(t)y(t) dt = f(b).
a
Let us express g(x)y from (3) via y and f and substitute the result into (4). Integrating
xx xx
by parts yields the desired boundary conditions for y(x),
sin[λ(b – a)]ϕ (b) – λ cos[λ(b – a)]ϕ(b)= λϕ(a),

x
(5)

sin[λ(b – a)]ϕ (a)+ λ cos[λ(b – a)]ϕ(a)= –λϕ(b); ϕ(x)= y(x) – f(x).
x
Equation (3) under the boundary conditions (5) determines the solution of the original
integral equation. Conditions (5) make it possible to calculate the constants of integration
that occur in solving the differential equation (3).
b
4.9-4. Equations of the Form y(x)+ K(x, t)y(···) dt = F (x)
a
b

40. y(x)+ f(t)y(x – t) dt =0.
a
Eigenfunctions of this integral equation* are determined by the roots of the following char-
acteristic (transcendental or algebraic) equation for µ:

b
f(t) exp(–µt) dt = –1. (1)
a
◦
1 . For a real (simple) root µ k of equation (1), there is a corresponding eigenfunction
y k (x)=exp(µ k x).
2 . For a real root µ k of multiplicity r, there are corresponding r eigenfunctions
◦
y k1 (x) = exp(µ k x), y k2 (x)= x exp(µ k x), ... , y kr (x)= x r–1 exp(µ k x).
3 . For a complex (simple) root µ k = α k + iβ k of equation (1), there is a corresponding pair
◦
of eigenfunctions
(2)
(1)
y (x) = exp(α k x) cos(β k x), y (x)=exp(α k x) sin(β k x).
k k
4 . For a complex root µ k = α k + iβ k of multiplicity r, there are corresponding r pairs of
◦
eigenfunctions
(1) (2)
y (x)=exp(α k x) cos(β k x), y (x) = exp(α k x) sin(β k x),
k1 k1
(1) (2)
y (x)= x exp(α k x) cos(β k x), y (x)= x exp(α k x) sin(β k x),
k2 k2
⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
(1) r–1 (2) r–1
y (x)= x exp(α k x) cos(β k x), y (x)= x exp(α k x) sin(β k x).
kr kr
The general solution is the linear combination (with arbitrary constants) of the eigenfunc-
tions of the homogeneous integral equation.
* In the equations below that contain y(x – t) in the integrand, the arguments can have, for example, the domain
(a) –∞ < x < ∞, –∞ < t < ∞ for a = –∞ and b = ∞ or (b) a ≤ t ≤ b, –∞ ≤ x < ∞, for a and b such that –∞ < a < b < ∞.
Case (b) is a special case of (a) if f(t) is nonzero only on the interval a ≤ t ≤ b.

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 325