Page 141 - Handbook Of Integral Equations

x

λ µ
51. y(x)+ A (x – t)x t y(t) dt = f(x).
a
–λ
The substitution u(x)= x y(x) leads to an equation of the form 2.1.49:
x
–λ
u(x)+ A (x – t)t λ+µ u(t) dt = f(x)x .
a
x

λ
λ
52. y(x)+ A (x – t )y(t) dt = f(x).
a
λ
This is a special case of equation 2.9.5 with g(x)= Ax .
Solution: x
1

y(x)= f(x)+ u (x)u (t) – u (x)u (t) f(t) dt,
1
2
2
1
W
a
where the primes denote differentiation with respect to the argument specified in the paren-
theses, and u 1 (x), u 2 (x) is a fundamental system of solutions of the second-order linear ho-
mogeneous ordinary differential equation u + Aλx λ–1 u = 0; the functions u 1 (x) and u 2 (x)
xx
are expressed in terms of Bessel functions or modified Bessel functions, depending on the
sign of A:
For Aλ >0,
√ √
2q √ Aλ q √ Aλ q λ +1
W = , u 1 (x)= xJ 1 x , u 2 (x)= xY 1 x , q = ,
π 2q q 2q q 2
For Aλ <0,
√ √
√ |Aλ| q √ |Aλ| q λ +1
W = –q, u 1 (x)= xI 1 x , u 2 (x)= xλK 1 x , q = .
2q q 2q q 2
x

λ λ–1 2λ–1
53. y(x) – Ax t + Bt y(t) dt = f(x).
a
The transformation
λ
λ
z = x , τ = t , y(x)= Y (z)
leads to an equation of the form 2.1.6:
z
A B
λ
Y (z) – z + τ Y (τ) dτ = F(z), F(z)= f(x), b = a .
b λ λ
x

λ+µ λ–µ–1 µ 2λ–µ–1
54. y(x) – Ax t + Bx t y(t) dt = f(x).
a
µ
The substitution y(x)= x w(x) leads to an equation of the form 2.1.53:
x
λ λ–1 2λ–1 –µ
w(x) – Ax t + Bt w(t) dt = x f(x).
a
x
λ–1 µ λ+µ–1
55. y(x)+ A λx t – (λ + µ)x y(t) dt = f(x).
a
This equation can be obtained by differentiating equation 1.1.51:
x x

λ µ λ+µ
1+ A(x t – x ) y(t) dt = F(x), F(x)= f(x) dx.
a a
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
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 120