Page 391 - Handbook Of Integral Equations

Chapter 6

Nonlinear Equations

With Constant Limits of Integration

Notation: f, g, h, and ϕ are arbitrary functions of an argument specified in the parentheses
(the argument can depend on t, x, and y); and A, B, C, a, b, c, s, β, γ, λ, and µ are arbitrary
parameters; and k, m, and n are nonnegative integers.

6.1. Equations With Quadratic Nonlinearity That Contain
Arbitrary Parameters

b

6.1-1. Equations of the Form K(t)y(x)y(t) dt = F (x)
a
1
λ
1. y(x)y(t) dt = Ax , A >0, λ > –1.
0
λ
This is a special case of equation 6.2.1 with f(x)= Ax , g(t)=1, a = 0, and b =1.
√
λ
Solutions: y(x)= ± A(λ +1) x .
1
2. y(x)y(t) dt = Ae βx , A >0.
0
βx
This is a special case of equation 6.2.1 with f(x)= Ae , g(t)=1, a = 0, and b =1.

Aβ βx
Solutions: y(x)= ± β e .
e –1
1
3. y(x)y(t) dt = A cosh(βx), A >0.
0
This is a special case of equation 6.2.1 with f(x)= A cosh(βx), g(t)=1, a = 0, and b =1.

Aβ
Solutions: y(x)= ± cosh(βx).
sinh β

1

4. y(x)y(t) dt = A sinh(βx), Aβ >0.
0
This is a special case of equation 6.2.1 with f(x)= A sinh(βx), g(t)=1, a = 0, and b =1.

Aβ
Solutions: y(x)= ± sinh(βx).
cosh β –1

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 371