Page 432 - Handbook Of Integral Equations

b

6.8-4. Equations of the Form y(x)+ G x, t, y(t) dt = F (x)
a
b

39. y(x)+ g 1 (x)f 1 t, y(t) + g 2 (x)f 2 t, y(t) dt = h(x).
a
A solution:
y(x)= h(x)+ λ 1 g 1 (x)+ λ 2 g 2 (x),

where the constants λ 1 and λ 2 are determined from the algebraic (or transcendental) system

b

λ 1 + f 1 t, h(t)+ λ 1 g 1 (t)+ λ 2 g 2 (t) dt =0,
a
b

λ 2 + f 2 t, h(t)+ λ 1 g 1 (t)+ λ 2 g 2 (t) dt =0.
a
n
b

40. y(x)+ g k (x)f k t, y(t) dt = h(x).
a
k=1
A solution:
n

y(x)= h(x)+ λ k g k (x),
k=1
where the coefficients λ k are determined from the algebraic (or transcendental) system

b n

λ m + f m t, h(t)+ λ k g k (t) dt =0; m =1, ... , n.
a
k=1
Different roots of this system generate different solutions of the integral equation.
•
Reference: A. F. Verlan’ and V. S. Sizikov (1986).

b
6.8-5. Equations of the Form F x, y(x) + G x, t, y(x), y(t) dt =0
a
b

41. y(x)+ 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 + λF(λ) – 1=0, F(λ)= f t, λg(t) dt.
a
b

42. y(x)+ g(x)y(x)f t, y(t) dt = h(x).
a
h(x)
A solution: y(x)= , where λ is determined from the algebraic (or transcendental)
1+ λg(x)
equation
b
h(t)
λ – F(λ)=0, F(λ)= f t, dt.
a 1+ λg(t)

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 412