Page 64 - Handbook Of Integral Equations

x

β λ β λ
20. t ln x – x ln t)y(t) dt = f(x).
a
λ
β
This is a special case of equation 1.9.11 with g(x)=ln x and h(t)= t .
x
β λ µ γ
21. At ln x + Bx ln t)y(t) dt = f(x).
a
λ
β
µ
This is a special case of equation 1.9.15 with g 1 (x)= A ln x, h 1 (t)= t , g 2 (x)= Bx , and
γ
h 2 (t)=ln t.
µ
x x + b
22. ln λ y(t) dt = f(x).
a ct + s
µ
λ
This is a special case of equation 1.9.6 with g(x) = ln(x + b) and h(t)= – ln(ct + s).
1.5. Equations Whose Kernels Contain Trigonometric
Functions

1.5-1. Kernels Containing Cosine

x
1. cos[λ(x – t)]y(t) dt = f(x).
a
x

Solution: y(x)= f (x)+ λ 2 f(x) dx.

x
a
x

2. cos[λ(x – t)] – 1 y(t) dt = f(x), f(a)= f (a)= f (x)=0.

x xx
a
1

Solution: y(x)= – f xxx (x) – f (x).
x
λ 2
x

3. cos[λ(x – t)] + b y(t) dt = f(x).
a
For b = 0, see equation 1.5.1. For b = –1, see equation 1.5.2. For λ = 0, see equation 1.1.1.
Differentiating the equation with respect to x, we arrive at an equation of the form 2.5.16:
λ x f (x)

x
y(x) – sin[λ(x – t)]y(t) dt = .
b +1 a b +1
1 . Solution with b(b + 1)>0:
◦
f (x) λ 2 x b

x

y(x)= + sin[k(x – t)]f (t) dt, where k = λ .
t
b +1 k(b +1) 2 a b +1
◦
2 . Solution with b(b + 1)<0:
f (x) λ 2 x –b

x

y(x)= + sinh[k(x – t)]f (t) dt, where k = λ .
t
b +1 k(b +1) 2 b +1
a
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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