Page 296 - Handbook Of Integral Equations

4.3-2. Kernels Containing Hyperbolic Sine

b
14. y(x) – λ sinh(βx)y(t) dt = f(x).
a
This is a special case of equation 4.9.1 with g(x) = sinh(βx) and h(t)=1.

b
15. y(x) – λ sinh(βt)y(t) dt = f(x).
a
This is a special case of equation 4.9.1 with g(x) = 1 and h(t) = sinh(βt).

b
16. y(x) – λ sinh[β(x – t)]y(t) dt = f(x).
a
This is a special case of equation 4.9.16 with g(x) = sinh(βx) and h(t) = cosh(βt).
Solution:

y(x)= f(x)+ λ A 1 sinh(βx)+ A 2 cosh(βx) ,
where A 1 and A 2 are the constants determined by the formulas presented in 4.9.16.

b
17. y(x) – λ sinh[β(x + t)]y(t) dt = f(x).
a
This is a special case of equation 4.9.15 with g(x) = sinh(βx) and h(t) = cosh(βt).
Solution:

y(x)= f(x)+ λ A 1 sinh(βx)+ A 2 cosh(βx) ,
where A 1 and A 2 are the constants determined by the formulas presented in 4.9.15.
n
b

18. y(x) – λ A k sinh[β k (x – t)] y(t) dt = f(x), n =1, 2, ...
a k=1
This is a special case of equation 4.9.20.

b sinh(βx)
19. y(x) – λ y(t) dt = f(x).
a sinh(βt)
1
This is a special case of equation 4.9.1 with g(x) = sinh(βx) and h(t)= .
sinh(βt)

b sinh(βt)
20. y(x) – λ y(t) dt = f(x).
a sinh(βx)
1
This is a special case of equation 4.9.1 with g(x)= and h(t) = sinh (βt).
sinh(βx)
b
m
k
21. y(x) – λ sinh (βx) sinh (µt)y(t) dt = f(x).
a
m
k
This is a special case of equation 4.9.1 with g(x) = sinh (βx) and h(t) = sinh (µt).
b
m
k
22. y(x) – λ t sinh (βx)y(t) dt = f(x).
a
m k
This is a special case of equation 4.9.1 with g(x) = sinh (βx) and h(t)= t .

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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