Page 80 - Handbook Of Integral Equations

x

β γ
104. A sin (λx)+ B cos (µt) y(t) dt = f(x).
a
γ
β
This is a special case of equation 1.9.6 with g(x)= A sin (λx) and h(t)= B cos (µt).
x
λ µ β γ
105. Ax cos t + Bt sin x y(t) dt = f(x).
a
λ
µ
γ
This is a special case of equation 1.9.15 with g 1 (x)= Ax , h 1 (t) = cos t, g 2 (x)= B sin x,
β
and h 2 (t)= t .
x
λ µ β γ
106. Ax sin t + Bt cos x y(t) dt = f(x).
a
µ
λ
γ
This is a special case of equation 1.9.15 with g 1 (x)= Ax , h 1 (t) = sin t, g 2 (x)= B cos x,
β
and h 2 (t)= t .
x

2
107. (x – t) sin[λ(x – t)] – λ(x – t) cos[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 5/2
g(t)= + λ (t – τ) J 5/2 [λ(t – τ)] f(τ) dτ.
2λ 64λ 5 dt 2 a
x sin[λ(x – t)]
108. – λ cos[λ(x – t)] y(t) dt = f(x).
a x – t
Solution:
2 3 x
1 d 2
y(x)= + λ sin[λ(x – t)]f(t) dt.
2λ 4 dx 2 a
x √ √ √

109. sin λ x – t – λ x – t cos λ x – t y(t) dt = f(x), f(a)= f (a)=0.

x
a
Solution: √
4 d 3 x cosh λ x – t
y(x)= √ f(t) dt.
3
πλ dx 3 a x – t
x

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

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

112. tan(λx) cot(µt) + tan(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) = cot(µt), g 2 (x) = tan(βx),
and h 2 (t) = cot(γt).
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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