Page 388 - Handbook Of Integral Equations

x
5.8-3. Equations of the Form y(x)+ K(x, t)G t, y(t) dt = F (x)
a
x

18. y(x)+ f t, y(t) dt = g(x).
a
The solution of this integral equation is determined by the solution of the first-order ordinary
differential equation

y + f(x, y) – g (x)=0
x x
under the initial condition y(a)= g(a). For the exact solutions of the first-order differential
equations with various f(x, y) and g(x), see E. Kamke (1977), A. D. Polyanin and V. F. Zaitsev
(1995), and V. F. Zaitsev and A. D. Polyanin (1994).
x

19. y(x)+ (x – t)f t, y(t) dt = g(x).
a
Differentiating the equation with respect to x yields
x

y + f t, y(t) dt = g (x). (1)

x x
a
In turn, differentiating this equation with respect to x yields the second-order nonlinear
ordinary differential equation
y + f(x, y) – g (x) = 0. (2)

xx xx
By setting x = a in the original equation and equation (1), we obtain the initial conditions for
y = y(x):

y(a)= g(a), y (a)= g (a). (3)

x
x
Equation (2) under conditions (3) defines the solution of the original integral equation. For
the exact solutions of the second-order differential equation (2) with various f(x, y) and g(x),
see A. D. Polyanin and V. F. Zaitsev (1995), and V. F. Zaitsev and A. D. Polyanin (1994).
x
n

20. y(x)+ (x – t) f t, y(t) dt = g(x), n =1, 2, ...
a
Differentiating the equation n+1 times with respect to x, we obtain an (n+1)st-order nonlinear
ordinary differential equation for y = y(x):
y (n+1) + n! f(x, y) – g (n+1) (x)=0.
x x
This equation under the initial conditions

y(a)= g(a), y (a)= g (a), ... , y (n) (a)= g x (n) (a),

x
x
x
defines the solution of the original integral equation.
x

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

x
x
a
Eliminating the integral term herefrom with the aid of the original equation, we obtain the
first-order nonlinear ordinary differential equation
y + f(x, y) – λy + λg(x) – g (x)=0.

x x
The unknown function y = y(x) must satisfy the initial condition y(a)= g(a). For the exact
solutions of the first-order differential equations with various f(x, y) and g(x), see E. Kamke
(1977), A. D. Polyanin and V. F. Zaitsev (1995), and V. F. Zaitsev and A. D. Polyanin (1994).
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 368