Page 405 - Handbook Of Integral Equations

b
n m
6.2-3. Equations of the Form y(x)+ K nm (x, t)y (x)y (t) dt = F (x), n+m ≤ 2
a
b

25. y(x)+ g(t)y(x)y(t) dt = f(x).
a
Solutions:
y 1 (x)= λ 1 f(x), y 2 (x)= λ 2 f(x),
where λ 1 and λ 2 are the roots of the quadratic equation
b
2
Iλ + λ – 1=0, I = f(t)g(t) dt.
a
b

26. y(x)+ g(x)y(x)y(t) dt = f(x).
a
A solution:
f(x)
y(x)= ,
1+ λg(x)
where λ is a root of the algebraic (or transcendental) equation
b
f(t) dt
λ – =0.
a 1+ λg(t)
Different roots generate different solutions of the integral equation.
b

2
27. y(x)+ g 1 (t)y (x)+ g 2 (x)y(t) dt = f(x).
a
Solution in an implicit form:
b
2
y(x)+ Iy (x)+ λg 2 (x) – f(x)=0, I = g 1 (t) dt, (1)
a
where λ is determined by the algebraic equation
b
λ = y(t) dt. (2)
a
Here the function y(x)= y(x, λ) obtained by solving the quadratic equation (1) must be
substituted in the integrand of (2).
b

2 2
28. y(x)+ g 1 (t)y (x)+ g 2 (x)y (t) dt = f(x).
a
Solution in an implicit form:
b

2
y(x)+ Iy (x)+ λg 2 (x) – f(x)=0, I = g 1 (t) dt, (1)
a
where λ is determined by the algebraic equation
b

2
λ = y (t) dt. (2)
a
Here the function y(x)= y(x, λ) obtained by solving the quadratic equation (1) must be
substituted into the integrand of (2).
b
2 2
29. y(x)+ g 11 (x)h 11 (t)y (x)+ g 12 (x)h 12 (t)y(x)y(t)+ g 22 (x)h 22 (t)y (t)
a

+ g 1 (x)h 1 (t)y(x)+ g 2 (x)h 2 (t)y(t) dt = f(x).
This is a special case of equation 6.8.44.

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 385