Page 35 - Handbook Of Integral Equations

x
1
47. b + y(t) dt = f(x), 0 < λ <1.
(x – t) λ
a
Rewrite the equation in the form
x x
y(t) dt
= f(x) – b y(t) dt.
(x – t) λ
a a
Assuming the right-hand side to be known, we solve this equation as the generalized Abel
equation 1.1.46. After some manipulations, we arrive at Abel’s equation of the second
kind 2.1.60:
b sin(πλ) x y(t) dt sin(πλ) d x f(t) dt
y(x)+ = F(x), where F(x)= .
π (x – t) 1–λ π dx (x – t) 1–λ
a a
x

√ √
λ
48. x – t y(t) dt = f(x), 0 < λ <1.
a
Solution:
2 x
k √ d f(t) dt sin(πλ)

y(x)= √ x √ √ √
, k = .
λ
x dx a t x – t πλ
x y(t) dt
49. √ √
λ = f(x), 0 < λ <1.
a x – t
Solution:
sin(πλ) d x f(t) dt
y(x)= √ √ 1–λ .
2π dx √
a t x – t
x
λ µ
50. Ax + Bt y(t) dt = f(x).
a
µ
λ
This is a special case of equation 1.9.6 with g(x)= Ax and h(t)= Bt .
x

λ µ λ+µ
51. 1+ A(x t – x ) y(t) dt = f(x).
a
µ
λ
This is a special case of equation 1.9.13 with g(x)= Ax and h(x)= x .
Solution:
d x λ x –λ Aµ µ+λ
y(x)= t f(t) Φ(t) dt , Φ(x)=exp – x .
dx Φ(x) a t µ + λ
x

β γ δ λ
52. Ax t + Bx t y(t) dt = f(x).
a
γ
β
δ
This is a special case of equation 1.9.15 with g 1 (x)= Ax , h 1 (t)= t , g 2 (x)= Bx , and
λ
h 2 (t)= t .
x

λ µ µ β γ γ
53. Ax (t – x )+ Bx (t – x ) y(t) dt = f(x).
a
µ
β
λ
This is a special case of equation 1.9.45 with g 1 (x)= Ax , h 1 (x)= x , g 2 (x)= Bx , and
γ
h 2 (x)= x .
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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