Page 423 - Handbook Of Integral Equations

6.8. Equations With Nonlinearity of General Form

b

6.8-1. Equations of the Form G(···) dt = F (x)
a
b

1. y(x)f t, y(t) dt = g(x).
a
A solution: y(x)= λg(x), where λ is determined by the algebraic (or transcendental) equation

b

λ f t, λg(t) dt =1.
a
b
k

2. y (x)f t, y(t) dt = g(x).
a
A solution: y(x)= λ[g(x)] 1/k , where λ is determined from the algebraic (or transcendental)
b
equation λ k f t, λg 1/k (t) dt =1.
a
b

3. ϕ y(x) f t, y(t) dt = g(x).
a
A solution in an implicit form:

λϕ y(x) – g(x) = 0, (1)
where λ is determined by the algebraic (or transcendental) equation
b

λ – F(λ)=0, F(λ)= f t, y(t) dt. (2)
a
Here the function y(x)= y(x, λ) obtained by solving (1) must be substituted into (2).
The number of solutions of the integral equation is determined by the number of the
solutions obtained from (1) and (2).

b

4. y(xt)f t, y(t) dt = A.
a
1 . Solutions: y(x)= λ k , where λ k are roots of the algebraic (or transcendental) equation
◦
b

λ f(t, λ) dt = A.
a
2 . Solutions: y(x)= px + q, where p and q are roots of the following system of algebraic
◦
(or transcendental) equations:
b b

tf(t, pt + q) dt =0, q f(t, pt + q) dt = A.
a a
¯

In the case f t, y(t) = f(t)y(t), see 6.2.2 for solutions of this system.
2 . 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 (or transcendental) equations.
◦
4 . The integral equation can have logarithmic solutions similar to those presented in item 3 ◦
of equation 6.2.2.

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 403