Page 59 - Handbook Of Integral Equations

x

99. A cosh(λx) sinh(µt)+ B cosh(βx) sinh(γt) y(t) dt = f(x).
a
This is a special case of equation 1.9.15 with g 1 (x)= A cosh(λx), h 1 (t) = sinh(µt), g 2 (x)=
B cosh(βx), and h 2 (t) = sinh(γt).
x

100. sinh(λx) cosh(µt) + sinh(βx) cosh(γt) y(t) dt = f(x).
a
This is a special case of equation 1.9.15 with g 1 (x) = sinh(λx), h 1 (t) = cosh(µt), g 2 (x)=
sinh(βx), and h 2 (t) = cosh(γt).

x

101. cosh(λx) cosh(µt) + sinh(βx) sinh(γt) y(t) dt = f(x).
a
This is a special case of equation 1.9.15 with g 1 (x) = cosh(λx), h 1 (t) = cosh(µt), g 2 (x)=
sinh(βx), and h 2 (t) = sinh(γt).
x

β γ
102. A cosh (λx)+ B sinh (µt) y(t) dt = f(x).
a
β
γ
This is a special case of equation 1.9.6 with g(x)= A cosh (λx) and h(t)= B sinh (µt).
x
β γ
103. A sinh (λx)+ B cosh (µt) y(t) dt = f(x).
a
β γ
This is a special case of equation 1.9.6 with g(x)= A sinh (λx) and h(t)= B cosh (µt).
x
λ µ β γ
104. Ax cosh t + Bt sinh 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) = cosh t, g 2 (x)= B sinh x,
β
and h 2 (t)= t .
x

2
105. (x – t) sinh[λ(x – t)] – λ(x – t) cosh[λ(x – t)] y(t) dt = f(x).
a
Solution:
x
y(x)= g(t) dt,
a
where
2 6 t
π 1 d 2 5
g(t)= – λ (t – τ) 2 I 5 [λ(t – τ)] f(τ) dτ.
2λ 64λ 5 dt 2 a 2
x
sinh[λ(x – t)]
106. – λ cosh[λ(x – t)] y(t) dt = f(x).
x – t
a
Solution:
2 3 x
1 d 2
y(x)= – λ sinh[λ(x – t)]f(t) dt.
2λ 4 dx 2 a
x √ √ √

107. sinh λ x – t – λ x – t cosh λ x – t y(t) dt = f(x), f(a)= f (a)=0.

x
a
Solution: √
4 d 3 x cos λ x – t
y(x)= – √ f(t) dt.
πλ dx 3 x – t
3
a
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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