Page 81 - Handbook Of Integral Equations

x

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

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

x

β γ
115. A tan (λx)+ B cot (µt) y(t) dt = f(x).
a
γ
β
This is a special case of equation 1.9.6 with g(x)= A tan (λx) and h(t)= B cot (µt).
x
β γ
116. A cot (λx)+ B tan (µt) y(t) dt = f(x).
a
γ
β
This is a special case of equation 1.9.6 with g(x)= A cot (λx) and h(t)= B tan (µt).
x

λ µ β γ
117. Ax tan t + Bt cot x y(t) dt = f(x).
a
γ
µ
λ
This is a special case of equation 1.9.15 with g 1 (x)= Ax , h 1 (t) = tan t, g 2 (x)= B cot x,
β
and h 2 (t)= t .
x

λ µ β γ
118. Ax cot t + Bt tan 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) = cot t, g 2 (x)= B tan x,
β
and h 2 (t)= t .
1.6. Equations Whose Kernels Contain Inverse
Trigonometric Functions
1.6-1. Kernels Containing Arccosine
x

1. arccos(λx) – arccos(λt) y(t) dt = f(x).
a
This is a special case of equation 1.9.2 with g(x) = arccos(λx).
1 d √

2 2
Solution: y(x)= – 1 – λ x f (x) .
x
λ dx
x

2. A arccos(λx)+ B arccos(λt) y(t) dt = f(x).
a
For B = –A, see equation 1.6.1. This is a special case of equation 1.9.4 with g(x) = arccos(λx).
Solution:
1 d – A x – B
y(x)= arccos(λx) A+B arccos(λt) A+B f (t) dt .

t
A + B dx a

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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