Page 78 - Handbook Of Integral Equations

x

µ µ
90. A cot x + B cot t y(t) dt = f(x).
a
µ
For B = –A, see equation 1.5.89. This is a special case of equation 1.9.4 with g(x) = cot x.
Solution:
1 d Aµ x Bµ
y(x)= tan x A+B tan t A+B f (t) dt .

t
A + B dx a
x

β γ
91. Ax + B cot (λt)+ C]y(t) dt = f(x).
a
β
γ
This is a special case of equation 1.9.6 with g(x)= Ax and h(t)= B cot (λt)+ C.
x

γ β
92. A cot (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 cot (λx) and h(t)= Bt + C.
x
λ µ β γ
93. Ax cot t + Bt cot 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 cot x,
β
and h 2 (t)= t .
1.5-5. Kernels Containing Combinations of Trigonometric Functions

x

94. cos[λ(x – t)] + A sin[µ(x – t)] y(t) dt = f(x).
a
Differentiating the equation with respect to x followed by eliminating the integral with the
cosine yields an equation of the form 2.3.16:

x

2
y(x) – (λ + A µ) sin[µ(x – t)] y(t) dt = f (x) – Aµf(x).

x
a
x

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

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

x

2 2
97. A cos (λx)+ B sin (µt) y(t) dt = f(x).
a
2
2
This is a special case of equation 1.9.6 with g(x)= A cos (λx) and h(t)= B sin (µt).

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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