Page 163 - Handbook Of Integral Equations

x
tanh(λx)
33. y(x) – A y(t) dt = f(x).
tanh(λt)
a
Solution:
x tanh(λx)
y(x)= f(x)+ A e A(x–t) f(t) dt.
a tanh(λt)
x tanh(λt)
34. y(x) – A y(t) dt = f(x).
a tanh(λx)
Solution:
x tanh(λt)
y(x)= f(x)+ A e A(x–t) f(t) dt.
a tanh(λx)
x

m
k
35. y(x) – A tanh (λx) tanh (µt)y(t) dt = f(x).
a
k
m
This is a special case of equation 2.9.2 with g(x)= A tanh (λx) and h(t) = tanh (µt).
x

k
m
36. y(x)+ A t tanh (λx)y(t) dt = f(x).
a
m
k
This is a special case of equation 2.9.2 with g(x)= –A tanh (λx) and h(t)= t .
x
m
k
37. y(x)+ A x tanh (λt)y(t) dt = f(x).
a
m
k
This is a special case of equation 2.9.2 with g(x)= –Ax and h(t) = tanh (λt).
∞

38. y(x)+ A tanh[λ(t – x)]y(t) dt = f(x).
x
This is a special case of equation 2.9.62 with K(z)= A tanh(–λz).
∞
√

39. y(x)+ A tanh λ t – x y(t) dt = f(x).
x
√
This is a special case of equation 2.9.62 with K(z)= A tanh λ –z .
x

40. y(x) – A tanh(kx)+ B – AB(x – t) tanh(kx) y(t) dt = f(x).
a
This is a special case of equation 2.9.7 with λ = B and g(x)= A tanh(kx).
x

41. y(x)+ A tanh(kt)+ B + AB(x – t) tanh(kt) y(t) dt = f(x).
a
This is a special case of equation 2.9.8 with λ = B and g(t)= A tanh(kt).
2.3-4. Kernels Containing Hyperbolic Cotangent
x

42. y(x) – A coth(λx)y(t) dt = f(x).
a
This is a special case of equation 2.9.2 with g(x)= A coth(λx) and h(t)=1.
Solution:
x sinh(λx)
A/λ
y(x)= f(x)+ A coth(λx) f(t) dt.
a sinh(λt)

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 142