Page 58 - Handbook Of Integral Equations

x

λ µ β γ
93. Ax coth t + Bt coth 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) = coth t, g 2 (x)= B coth x,
β
and h 2 (t)= t .
1.3-5. Kernels Containing Combinations of Hyperbolic Functions
x

94. cosh[λ(x – t)] + A sinh[µ(x – t)] y(t) dt = f(x).
a
Let us differentiate the equation with respect to x and then eliminate the integral with the
hyperbolic cosine. As a result, we arrive at an equation of the form 2.3.16:
x
2
y(x)+(λ – A µ) sinh[µ(x – t)]y(t) dt = f (x) – Aµf(x).

x
a
x

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

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

97. sinh[λ(x – t)] cosh[λ(x + t)]y(t) dt = f(x).
a
Using the formula

sinh(α – β) cosh(α + β)= 1 sinh(2α) – sinh(2β) , α = λx, β = λt,
2
we reduce the original equation to an equation of the form 1.3.37:
x

sinh(2λx) – sinh(2λt) y(t) dt =2f(x).
a

1 d f (x)
x
Solution: y(x)= .
λ dx cosh(2λx)
x
98. cosh[λ(x – t)] sinh[λ(x + t)]y(t) dt = f(x).
a
Using the formula

cosh(α – β) sinh(α + β)= 1 sinh(2α) + sinh(2β) , α = λx, β = λt,
2
we reduce the original equation to an equation of the form 1.3.38 with A = B =1:
x

sinh(2λx) + sinh(2λt) y(t) dt =2f(x).
a

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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