Page 398 - Handbook Of Integral Equations

∞ x

40. y(x)+ A e –λt y y(t) dt =0, λ >0.
0 t
λ C
A solution: y(x)= – x , where C is an arbitrary constant.
A
∞ x

b
41. y(x)+ A e –λt y y(t) dt = Bx , λ >0.
t
0
Solutions:
b
b
y 1 (x)= β 1 x , y 2 (x)= β 2 x ,
2
where β 1 and β 2 are the roots of the quadratic equation Aβ + λβ – Bλ =0.
6.2. Equations With Quadratic Nonlinearity That Contain
Arbitrary Functions

b

6.2-1. Equations of the Form G(···) dt = F (x)
a
b
1. g(t)y(x)y(t) dt = f(x).
a
Solutions:
b –1/2

y(x)= ±λf(x), λ = f(t)g(t) dt .
a
b
2. f(t)y(t)y(xt) dt = A.
a
1 . Solutions*
◦

y 1 (x)= A/I 0 , y 2 (x)= – A/I 0 ,
y 3 (x)= q(I 1 x – I 2 ), y 4 (x)= –q(I 1 x – I 2 ),
where
b
1/2
A
m
I m = t f(t) dt, q = , m =0, 1, 2.
2 2
a I 0 I – I I 2
2
1
The integral equation has some other (more complicated) solutions of the polynomial
k
form y(x)= n B k x , where the constants B k can be found from the corresponding system
k=0
of algebraic equations.
◦
2 . Solutions:
C
C
y 5 (x)= q(I 1 x – I 2 ), y 6 (x)= –q(I 1 x – I 2 ),
1/2
A mC

b
q = , I m = t f(t) dt, m =0, 1, 2,
2 2
I 0 I – I I 2 a
2 1
where C is an arbitrary constant.
The equation has more complicated solutions of the form y(x)= n B k x kC , where C is
k=0
an arbitrary constant and the coefficients B k can be found from the corresponding system of
algebraic equations.
* The arguments of the equations containing y(xt) in the integrand can vary, for example, within the following intervals:
(a) 0 ≤ t ≤ 1, 0 ≤ x ≤ 1 for a = 0 and b = 1; (b) 1 ≤ t < ∞,1 ≤ x < ∞ for a = 1 and b = ∞; (c) 0 ≤ t < ∞,0 ≤ x < ∞ for
a = 0 and b = ∞; or (d) a ≤ t ≤ b,0 ≤ x < ∞ for arbitrary a and b such that 0 ≤ a < b ≤ ∞. Case (d) is a special case of (c)
if f(t) is nonzero only on the interval a ≤ t ≤ b.
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 378