Page 76 - Handbook Of Integral Equations

x

√
73. tan x – tan ty(t) dt = f(x).
a
Solution: x
2 2 d
2 f(t) dt
y(x)= cos x √ .
2
π cos x dx a cos t tan x – tan t
2
x
y(t) dt

74. √ = f(x).
a tan x – tan t
Solution: x
1 d f(t) dt
y(x)= √ .
π dx a cos t tan x – tan t
2
x

λ
75. (tan x – tan t) y(t) dt = f(x), 0 < λ <1.
a
Solution:
sin(πλ) 2 d
2 x f(t) dt
y(x)= cos x .
πλ cos x dx cos t(tan x – tan t) λ
2
2
a
x
µ
µ
76. (tan x – tan t)y(t) dt = f(x).
a
µ
This is a special case of equation 1.9.2 with g(x) = tan x.
µ+1
1 d cos xf (x)

x
Solution: y(x)= .
µ dx sin µ–1 x
x

µ µ
77. A tan x + B tan t y(t) dt = f(x).
a
µ
For B = –A, see equation 1.5.76. This is a special case of equation 1.9.4 with g(x) = tan x.
Solution:
1 d – Aµ x – Bµ

y(x)= tan(λx) A+B tan(λt) A+B f (t) dt .
t
A + B dx a
x y(t) dt
78. = f(x), 0 < µ <1.
a [tan(λx) – tan(λt)] µ
This is a special case of equation 1.9.42 with g(x) = tan(λx) and h(x) ≡ 1.
Solution:
λ sin(πµ) d x f(t) dt
y(x)= .
2
π dx a cos (λt)[tan(λx) – tan(λt)] 1–µ
x
β γ
79. Ax + B tan (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 tan (λt)+ C.
x

γ β
80. A tan (λx)+ Bt + C]y(t) dt = f(x).
a
β
γ
This is a special case of equation 1.9.6 with g(x)= A tan (λx) and h(t)= Bt + C.
x
λ µ β γ
81. Ax tan 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) = tan t, g 2 (x)= B tan x,
β
and h 2 (t)= t .
© 1998 by CRC Press LLC
© 1998 by CRC Press LLC
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