Page 110 - Handbook Of Integral Equations

x

√
√
60. AI ν λ x + BI ν λ t y(t) dt = f(x).
a
Solution with B ≠ –A:
1 d √
– A x √
– B
y(x)= I ν λ x A+B I ν λ t A+B f (t) dt .

t
A + B dx a
x
√
√
61. AI ν λ x + BI µ β t y(t) dt = f(x).
a
√
√
This is a special case of equation 1.9.6 with g(x)= AI ν λ x and h(t)= BI µ β t .
x √ √

62. x – tI 1 λ x – t y(t) dt = f(x).
a
Solution: x
2 d 3 √
y(x)= J 0 λ x – t f(t) dt.
λ dx 3 a

x
√
63. (x – t) 1/4 I 1/2 λ x – t y(t) dt = f(x).
a
Solution: √
2 x
2 d cos λ x – t
y(x)= √ f(t) dt.
πλ dx 2 a x – t
x √

64. (x – t) 3/4 I 3/2 λ x – t y(t) dt = f(x).
a
Solution: √
2 3/2 d 3 x cos λ x – t
y(x)= √ 3 √ f(t) dt.
πλ 3/2 dx a x – t

x √

65. (x – t) n/2 I n λ x – t y(t) dt = f(x), n =0, 1, 2, ...
a
Solution:
2 d

n n+2 x √
y(x)= J 0 λ x – t f(t) dt.
λ dx n+2 a
x 2n–3 √

66. x – t 4 I 2n–3 λ x – t y(t) dt = f(x), n =1, 2, ...
a 2
Solution:
2n–3 x √
1 2 2 d n cos λ x – t
y(x)= √ √ f(t) dt.
π λ dx n a x – t
x
√
–1/4
67. (x – t) I –1/2 λ x – t y(t) dt = f(x).
a
Solution:
x √
λ d cos λ x – t
y(x)= √ f(t) dt.
2π dx x – t
a

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 88