Page 306 - Handbook Of Integral Equations

n
b

21. y(x) – λ A k sin[β k (x – t)] y(t) dt = f(x), n =1, 2, ...
a
k=1
This equation can be reduced to a special case of equation 4.9.20; the formula sin[β(x – t)] =
sin(βx) cos(βt) – sin(βt) cos(βx) must be used.
b sin(βx)
22. y(x) – λ y(t) dt = f(x).
a sin(βt)
1
This is a special case of equation 4.9.1 with g(x) = sin(βx) and h(t)= .
sin(βt)
b sin(βt)
23. y(x) – λ y(t) dt = f(x).
a sin(βx)
1
This is a special case of equation 4.9.1 with g(x)= and h(t) = sin (βt).
sin(βx)
b

k
m
24. y(x) – λ sin (βx) sin (µt)y(t) dt = f(x).
a
m
k
This is a special case of equation 4.9.1 with g(x) = sin (βx) and h(t) = sin (µt).
b
m
k
25. y(x) – λ t sin (βx)y(t) dt = f(x).
a
k
m
This is a special case of equation 4.9.1 with g(x) = sin (βx) and h(t)= t .

b
m
k
26. y(x) – λ x sin (βt)y(t) dt = f(x).
a
k
m
This is a special case of equation 4.9.1 with g(x)= x and h(t) = sin (βt).
b
27. y(x) – λ [A + B(x – t) sin(βt)]y(t) dt = f(x).
a
This is a special case of equation 4.9.8 with h(t) = sin(βt).
b
28. y(x) – λ [A + B(x – t) sin(βx)]y(t) dt = f(x).
a
This is a special case of equation 4.9.10 with h(x) = sin(βx).
b

29. y(x)+ A sin(λ|x – t|)y(t) dt = f(x).
a
This is a special case of equation 4.9.39 with g(t)= A.
◦
1 . The function y = y(x) obeys the following second-order linear nonhomogeneous ordinary
differential equation with constant coefficients:
2

y + λ(2A + λ)y = f (x)+ λ f(x). (1)
xx
xx
The boundary conditions for (1) have the form (see 4.9.39)

sin[λ(b – a)]ϕ (b) – λ cos[λ(b – a)]ϕ(b)= λϕ(a),
x
ϕ(x)= y(x) – f(x). (2)
sin[λ(b – a)]ϕ (a)+ λ cos[λ(b – a)]ϕ(a)= –λϕ(b),

x
Equation (1) under the boundary conditions (2) determines the solution of the original
integral equation.
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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