Page 56 - Handbook Of Integral Equations

x

µ
µ
76. (tanh x – tanh t)y(t) dt = f(x).
a
µ
This is a special case of equation 1.9.2 with g(x) = tanh x.
1 d cosh µ+1 xf (x)

x
Solution: y(x)= .
µ dx sinh µ–1 x
x
µ µ
77. A tanh x + B tanh t y(t) dt = f(x).
a
µ
For B = –A, see equation 1.3.76. This is a special case of equation 1.9.4 with g(x) = tanh x.
Solution:
Aµ x Bµ
1 d – –

y(x)= tanh(λx) A+B tanh(λt) A+B f (t) dt .
t
A + B dx a
x
y(t) dt

78. = f(x), 0 < µ <1.
a [tanh(λx) – tanh(λt)] µ
This is a special case of equation 1.9.42 with g(x) = tanh(λx) and h(x) ≡ 1.
Solution:
λ sin(πµ) d x f(t) dt
y(x)= .
2
π dx a cosh (λt)[tanh(λx) – tanh(λt)] 1–µ
x
β γ
79. Ax + B tanh (λt)+ C]y(t) dt = f(x).
a
β
γ
This is a special case of equation 1.9.6 with g(x)= Ax and h(t)= B tanh (λt)+ C.
x

γ β
80. A tanh (λx)+ Bt + C]y(t) dt = f(x).
a
γ β
This is a special case of equation 1.9.6 with g(x)= A tanh (λx) and h(t)= Bt + C.
x

λ µ β γ
81. Ax tanh t + Bt tanh x y(t) dt = f(x).
a
µ
γ
λ
This is a special case of equation 1.9.15 with g 1 (x)= Ax , h 1 (t) = tanh t, g 2 (x)= B tanh x,
β
and h 2 (t)= t .
1.3-4. Kernels Containing Hyperbolic Cotangent
x

82. coth(λx) – coth(λt) y(t) dt = f(x).
a
This is a special case of equation 1.9.2 with g(x) = coth(λx).
1 d 2
Solution: y(x)= – sinh (λx)f (x) .
x
λ dx
x

83. A coth(λx)+ B coth(λt) y(t) dt = f(x).
a
For B = –A, see equation 1.3.82. This is a special case of equation 1.9.4 with g(x) = coth(λx).
1 d A x B

Solution: y(x)= tanh(λx) A+B tanh(λt) A+B f (t) dt .
t
A + B dx a
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 34