Page 365 - Handbook Of Integral Equations

x
t
µ λx
5. f y(t)y(x – t) dt = Ax e .
x
0
Solutions:
1
A µ–1 λx µ–1 µ–1
y(x)= ± x 2 e , I = f(z)z 2 (1 – z) 2 dz.
I 0
x t
λ
6. f y(t)y(ax + bt) dt = Ax .
0 x
Solutions:
1
A λ–1 λ–1 λ–1
y(x)= ± x 2 , I = f(z)z 2 (a + bz) 2 dz.
I 0
x t
7. f y(t)y(ax – t) dt = Ae λx , a ≥ 1.
0 x
Solutions:
1
A exp(λx/a) f(z) dz
y(x)= ± √ , I = √ .
I x 0 z(a – z)
x
t
µ λx
8. f y(t)y(ax – t) dt = Ax e , a ≥ 1.
0 x
Solutions:
1
A µ–1 µ–1 µ–1
y(x)= ± x 2 exp(λx/a), I = f(z)z 2 (a – z) 2 dz.
I
0
x 2
5.2-2. Equations of the Form y(x)+ K(x, t)y (t) dt = F (x)
a
x

2
9. y(x)+ f(t)y (t) dt = A.
a
Solution:
x
–1
y(x)= A 1+ A f(t) dt .
a
x

2
10. y(x)+ e λ(x–t) g(t)y (t) dt = f(x).
a
Differentiating the equation with respect to x yields
x

2
2

y + g(x)y + λ e λ(x–t) g(t)y (t) dt = f (x). (1)
x x
a
Eliminating the integral term from (1) with the aid of the original equation, we arrive at a
Riccati ordinary differential equation,
2

y + g(x)y – λy + λf(x) – f (x)=0, (2)

x
x
under the initial condition y(a)= f(a). Equation (2) can be reduced to a second-order linear
ordinary differential equation. For the exact solutions of equation (2) with various specific
functions f and g, see, for example, E. Kamke (1977) and A. D. Polyanin and V. F. Zaitsev
(1995).
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 345