Page 142 - Handbook Of Integral Equations

x

λ+1 λ λ
56. y(x)+ ABx – ABx t – Ax – B y(t) dt = f(x).
a
This is a special case of equation 2.9.7.
Solution:
x
y(x)= f(x)+ R(x – t)f(t) dt,
a
x

λ
R(x, t)=(Ax +B)exp A x λ+1 –t λ+1 +B 2 exp A s λ+1 –t λ+1 +B(x–s) ds.
λ +1 t λ +1
x
λ λ+1 λ
57. y(x)+ ABxt – ABt + At + B y(t) dt = f(x).
a
This is a special case of equation 2.9.8.
Solution:
x

y(x)= f(x)+ R(x – t)f(t) dt,
a
x
λ
R(x, t)= –(At +B)exp A t λ+1 –x λ+1 +B 2 exp A s λ+1 –x λ+1 +B(t–s) ds.
λ +1 t λ +1

x
x + b µ
58. y(x) – λ y(t) dt = f(x).
a t + b
µ
This is a special case of equation 2.9.1 with g(x)=(x + b) .
Solution:
x + b λ(x–t)
µ
x
y(x)= f(x)+ λ e f(t) dt.
a t + b
µ
x x + b
59. y(x) – λ µ y(t) dt = f(x).
a t + b
µ
This is a special case of equation 2.9.1 with g(x)= x + b.
Solution:
x
µ
x + b
y(x)= f(x)+ λ µ e λ(x–t) f(t) dt.
a t + b
x
y(t) dt
60. y(x) – λ = f(x), 0 < α <1.
0 (x – t) α
Generalized Abel equation of the second kind.
◦
1 . Assume that the number α can be represented in the form
m
α =1 – , where m =1, 2, ... , n =2, 3, ... (m < n).
n
In this case, the solution of the generalized Abel equation of the second kind can be written
in closed form (in quadratures):
x

y(x)= f(x)+ R(x – t)f(t) dt,
0

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 121