Page 108 - Handbook Of Integral Equations

x

49. (x – t) 1/2 I 1/2 [λ(x – t)]y(t) dt = f(x).
a
Solution:
2 3 x
π d 2 3/2
y(x)= – λ (x – t) I 3/2 [λ(x – t)] f(t) dt.
4λ 2 dx 2 a
x

50. (x – t) 3/2 I 1/2 [λ(x – t)]y(t) dt = f(x).
a
Solution:
x

y(x)= g(t) dt,
a
where
π d 2 2 4 t 3/2
g(t)= – λ (t – τ) I 3/2 [λ(t – τ)] f(τ) dτ.
8λ 2 dt 2 a
x
51. (x – t) 3/2 I 3/2 [λ(x – t)]y(t) dt = f(x).
a
Solution:
√ 2 3 x
π d 2
y(x)= – λ sinh[λ(x – t)] f(t) dt.
λ
2 3/2 5/2 dx 2 a
x
52. (x – t) 5/2 I 3/2 [λ(x – t)]y(t) dt = f(x).
a
Solution:
x
y(x)= g(t) dt,
a
where
π d 2 2 6 t 5/2
g(t)= – λ (t – τ) I 5/2 [λ(t – τ)] f(τ) dτ.
128λ 4 dt 2 a

x
2n–1

53. x – t 2 I 2n–1 [λ(x – t)]y(t) dt = f(x), n =2, 3, ...
a 2
Solution:
√ 2 n x
π d 2
y(x)= – λ sinh[λ(x – t)] f(t) dt.
√ 2n+1 dx 2
2 λ 2 (2n – 2)!! a
x

54. [I ν (λx) – I ν (λt)]y(t) dt = f(x).
a
This is a special case of equation 1.9.2 with g(x)= I ν (λx).

x

55. [AI ν (λx)+ BI ν (λt)]y(t) dt = f(x).
a
Solution with B ≠ –A:
1 d – A x – B
y(x)= I ν (λx) A+B I ν (λt) A+B f (t) dt .

t
A + B dx a

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 86