Page 57 - Handbook Of Integral Equations

x

84. A coth(λx)+ B coth(µt)+ C y(t) dt = f(x).
a
This is a special case of equation 1.9.6 with g(x)= A coth(λx) and h(t)= B coth(µt)+ C.
x

2 2
85. coth (λx) – coth (λt) y(t) dt = f(x).
a
2
This is a special case of equation 1.9.2 with g(x) = coth (λx).
3
d sinh (λx)f (x)

x
Solution: y(x)= – .
dx 2λ cosh(λx)
x
2 2
86. A coth (λx)+ B coth (λt) y(t) dt = f(x).
a
2
For B = –A, see equation 1.3.85. This is a special case of equation 1.9.4 with g(x) = coth (λx).
1 d 2A x 2B
Solution: y(x)= tanh(λx) A+B tanh(λt) A+B f (t) dt .

t
A + B dx a
x

2 2
87. A coth (λx)+ B coth (µt)+ C y(t) dt = f(x).
a
2
2
This is a special case of equation 1.9.6 with g(x)= A coth (λx) and h(t)= B coth (µt)+ C.
x

n
88. coth(λx) – coth(λt) y(t) dt = f(x), n =1, 2, ...
a

The right-hand side of the equation is assumed to satisfy the conditions f(a)= f (a)= ··· =
x
f (n) (a)=0.
x n+1
(–1) n 2 d
Solution: y(x)= 2 sinh (λx) f(x).
n
λ n! sinh (λx) dx
x
µ µ
89. (coth x – coth t)y(t) dt = f(x).
a
µ
This is a special case of equation 1.9.2 with g(x) = coth x.
1 d sinh µ+1 xf (x)

x
Solution: y(x)= – .
µ dx cosh µ–1 x
x

µ µ
90. A coth x + B coth t y(t) dt = f(x).
a
µ
For B = –A, see equation 1.3.89. This is a special case of equation 1.9.4 with g(x) = coth x.
Solution:
1 d Aµ x Bµ

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

β γ
91. Ax + B coth (λt)+ C]y(t) dt = f(x).
a
γ
β
This is a special case of equation 1.9.6 with g(x)= Ax and h(t)= B coth (λt)+ C.
x
γ β
92. A coth (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 coth (λx) and h(t)= Bt + C.
© 1998 by CRC Press LLC
© 1998 by CRC Press LLC
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