Page 89 - Handbook Of Integral Equations

x e µ(x–t) y(t) dt
13. = f(x), 0 < λ <1.
a (cosh x – cosh t) λ
Solution:
sin(πλ) µx d x e –µt sinh tf(t) dt
y(x)= e .
π dx a (cosh x – cosh t) 1–λ
x

14. e µ(x–t) A 1 sinh[λ 1 (x – t)] + A 2 sinh[λ 2 (x – t)] y(t) dt = f(x).
a
The substitution w(x)= e –µx y(x) leads to an equation of the form 1.3.41:
x

–µx
A 1 sinh[λ 1 (x – t)] + A 2 sinh[λ 2 (x – t)] w(t) dt = e f(x).
a
x
2
15. e µ(x–t) sinh [λ(x – t)]y(t) dt = f(x).
a
The substitution w(x)= e –µx y(x) leads to an equation of the form 1.3.43:
x
2
sinh [λ(x – t)]w(t) dt = e –µx f(x).
a
x

3
16. e µ(x–t) sinh [λ(x – t)]y(t) dt = f(x).
a
The substitution w(x)= e –µx y(x) leads to an equation of the form 1.3.49:
x

3
sinh [λ(x – t)]w(t) dt = e –µx f(x).
a
x
n
17. e µ(x–t) sinh [λ(x – t)]y(t) dt = f(x), n =2, 3, ...
a
The substitution w(x)= e –µx y(x) leads to an equation of the form 1.3.54:
x

n
sinh [λ(x – t)]w(t) dt = e –µx f(x).
a
x
√
18. e µ(x–t) sinh k x – t y(t) dt = f(x).
a
Solution: √
2 µx d 2 x e –µt cos k x – t
y(x)= e √ f(t) dt.
πk dx 2 a x – t
x
√
19. e µ(x–t) sinh x – sinh ty(t) dt = f(x).
a
Solution:
2 µx 1 d
2 x e –µt cosh tf(t) dt
y(x)= e cosh x √ .
π cosh x dx a sinh x – sinh t
x µ(x–t)
e y(t) dt
20. √ = f(x).
a sinh x – sinh t
Solution: x
1 µx d e –µt cosh tf(t) dt
y(x)= e √ .
π dx sinh x – sinh t
a

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 67