Page 364 - Handbook Of Integral Equations

x

2
46. y(x)+ A y(t)y(x – t) dt =(AB x + B)e λx .
0
λx
A solution: y(x)= Be .
λ x 1
47. y(x)+ y(t)y(x – t) dt = 2 β sinh(λx).
2β 0
A solution: y(x)= βI 1 (λx), where I 1 (x) is the modified Bessel function.

λ x
48. y(x) – y(t)y(x – t) dt = 1 2 β sin(λx).
2β 0
A solution: y(x)= βJ 1 (λx), where J 1 (x) is the Bessel function.

x

λ
49. y(x)+ A x –λ–1 y(t)y(x – t) dt = Bx .
0
λ
λ
Solutions: y 1 (x)= β 1 x and y 2 (x)= β 2 x , where β 1,2 are the roots of the quadratic equation
2
1 Γ (λ +1)
2
λ
λ
AIβ + β – B =0, I = z (1 – z) dz = .
0 Γ(2λ +2)
5.2. Equations With Quadratic Nonlinearity That Contain
Arbitrary Functions
x
5.2-1. Equations of the Form G(···) dt = F (x)
a
x

2
1. K(x, t)y (t) dt = f(x).
a
2
The substitution w(x)= y (x) leads to the linear equation
x

K(x, t)w(t) dt = f(x).
a
x
2. K(t)y(x)y(t) dt = f(x).
a
Solutions:
x
–1/2

y(x)= ±f(x) 2 K(t)f(t) dt .
a
x
t
λ
3. f y(t)y(x – t) dt = Ax .
x
0
Solutions:
1
A λ–1 λ–1 λ–1
y(x)= ± x 2 , I = f(z)z 2 (1 – z) 2 dz.
I
0
x
t
4. f y(t)y(x – t) dt = Ae λx .
x
0
Solutions:
λx 1
A e f(z) dz
y(x)= ± √ , I = √ .
I x z(1 – z)
0
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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