Page 79 - Handbook Of Integral Equations

x

98. sin[λ(x – t)] cos[λ(x + t)]y(t) dt = f(x), f(a)= f (a)=0.
x
a
Using the trigonometric formula

sin(α – β) cos(α + β)= 1 sin(2α) – sin(2β) , α = λx, β = λt,
2
we reduce the original equation to an equation of the form 1.5.37:

x

sin(2λx) – sin(2λt) y(t) dt =2f(x).
a
1 d f (x)

x
Solution: y(x)= .
λ dx cos(2λx)
x

99. cos[λ(x – t)] sin[λ(x + t)]y(t) dt = f(x).
a
Using the trigonometric formula

cos(α – β) sin(α + β)= 1 sin(2α) + sin(2β) , α = λx, β = λt,
2
we reduce the original equation to an equation of the form 1.5.38 with A = B =1:

x

sin(2λx) + sin(2λt) y(t) dt =2f(x).
a
Solution with sin(2λx)>0:

d 1 x f (t) dt
t
y(x)= √ √ .
dx sin(2λx) a sin(2λt)
x

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

x

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

102. A cos(λx) cos(µt)+ B sin(βx) sin(γt) y(t) dt = f(x).
a
This is a special case of equation 1.9.15 with g 1 (x)= A cos(λx), h 1 (t) = cos(µt), g 2 (x)=
B sin(βx), and h 2 (t) = sin(γt).
x
β γ
103. A cos (λx)+ B sin (µt) y(t) dt = f(x).
a
γ
β
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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