Page 47 - Handbook Of Integral Equations

x
√
21. cosh x – cosh ty(t) dt = f(x).
a
Solution:
2 1 d
2 x sinh tf(t) dt
y(x)= sinh x √ .
π sinh x dx
a cosh x – cosh t
x
y(t) dt
22. √ = f(x).
a cosh x – cosh t
Solution: x
1 d sinh tf(t) dt
y(x)= √ .
π dx a cosh x – cosh t
x

λ
23. (cosh x – cosh t) y(t) dt = f(x), 0 < λ <1.
a
Solution:
1 d sinh tf(t) dt sin(πλ)

2 x
y(x)= k sinh x , k = .
sinh x dx a (cosh x – cosh t) λ πλ
x

µ
µ
24. (cosh x – cosh t)y(t) dt = f(x).
a
µ
This is a special case of equation 1.9.2 with g(x) = cosh x.

1 d f (x)

x
Solution: y(x)= µ–1 .
µ dx sinh x cosh x
x
µ µ
25. A cosh x + B cosh t y(t) dt = f(x).
a
µ
For B = –A, see equation 1.3.24. This is a special case of equation 1.9.4 with g(x) = cosh x.
Solution:
Aµ x Bµ
1 d – –

y(x)= cosh(λx) A+B cosh(λt) A+B f (t) dt .
t
A + B dx
a
x
y(t) dt
26. = f(x), 0 < λ <1.
a (cosh x – cosh t) λ
Solution: x
sin(πλ) d sinh tf(t) dt
y(x)= .
π dx a (cosh x – cosh t) 1–λ
x
27. (x – t) cosh[λ(x – t)]y(t) dt = f(x), f(a)= f (a)=0.

x
a
Differentiating the equation twice yields
x x

y(x)+2λ sinh[λ(x – t)]y(t) dt + λ 2 (x – t) cosh[λ(x – t)]y(t) dt = f (x).
xx
a a
Eliminating the third term on the right-hand side with the aid of the original equation, we
arrive at an equation of the form 2.3.16:
x
2

y(x)+2λ sinh[λ(x – t)]y(t) dt = f (x) – λ f(x).
xx
a
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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