Page 412 - Handbook Of Integral Equations

b

6.3-3. Equations of the Form y(x)+ G(···) dt = F (x)
a
b

µ
β
16. y(x)+ A y (x)y (t) dt = f(x).
a
Solution in an implicit form:
β
y(x)+ Aλy (x) – f(x)=0, (1)
where λ is determined by the algebraic (or transcendental) equation
b

µ
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
µ
17. y(x)+ g(t)y(x)y (t) dt = f(x).
a
A solution: y(x)= λf(x), where λ is determined from the algebraic (or transcendental)
equation
b
µ
Iλ µ+1 + λ – 1=0, I = g(t)f (t) dt.
a
b

µ
18. 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
µ
b f (t) dt
λ – µ =0.
a [1 + λg(t)]
Different roots generate different solutions of the integral equation.
b
2 µ
19. 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 (or transcendental) equation
b

µ
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
k s p q
20. y(x)+ g 1 (x)h 1 (t)y (x)y (t)+ g 2 (x)h 2 (t)y (x)y (t) dt = f(x).
a
This is a special case of equation 6.8.44.
b
β
21. y(x)+ A y(xt)y (t) dt =0.
a
β
This is a special case of equation 6.8.45 with f(t, y)= Ay .

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 392