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x

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

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

x
Solution: y(x)= .
dx 2λ sinh(λx)
x

2 2
70. A tanh (λx)+ B tanh (λt) y(t) dt = f(x).
a
2
For B = –A, see equation 1.3.69. This is a special case of equation 1.9.4 with g(x) = tanh (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
71. A tanh (λx)+ B tanh (µt)+ C y(t) dt = f(x).
a
2 2
This is a special case of equation 1.9.6 with g(x)= A tanh (λx) and h(t)= B tanh (µt)+ C.
x
n
72. tanh(λx) – tanh(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 2 d
Solution: y(x)= 2 cosh (λx) f(x).
n
λ n! cosh (λx) dx
x

√
73. tanh x – tanh ty(t) dt = f(x).
a
Solution:
2 2 d
2 x f(t) dt
y(x)= 2 cosh x 2 √ .
π cosh x dx a cosh t tanh x – tanh t
x
y(t) dt
74. √ = f(x).
a tanh x – tanh t
Solution: x
1 d f(t) dt
y(x)= √ .
2
π dx a cosh t tanh x – tanh t
x

λ
75. (tanh x – tanh t) y(t) dt = f(x), 0 < λ <1.
a
Solution:
sin(πλ) 2 d
2 x f(t) dt
y(x)= 2 cosh x 2 .
πλ cosh x dx a cosh t (tanh x – tanh t) λ
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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