Page 62 - Handbook Of Integral Equations

x

2 2

6. ln (λx) – ln (λt) y(t) dt = f(x), f(a)= f (a)=0.
x
a

d xf (x)

x
Solution: y(x)= .
dx 2 ln(λx)
x

2 2
7. A ln (λx)+ B ln (λt) y(t) dt = f(x).
a
2
For B = –A, see equation 1.4.7. This is a special case of equation 1.9.4 with g(x)=ln (λx).
Solution:
1 d – 2A x – 2B

y(x)= ln(λx) A+B ln(λt) A+B f (t) dt .
t
A + B dx
a
x
2 2
8. A ln (λx)+ B ln (µt)+ C y(t) dt = f(x).
a
2
2
This is a special case of equation 1.9.6 with g(x)=ln (λx) and h(t)=ln (µt)+ C.
x

n
9. ln(x/t) y(t) dt = f(x), n =1, 2, ...
a
The right-hand side of the equation is assumed to satisfy the conditions f(a)= f (a)= ··· =

x
f (n) (a)=0.
x
n+1
1 d
Solution: y(x)= x f(x).
n! x dx
x

2 2
n
10. ln x – ln t y(t) dt = f(x), n =1, 2, ...
a
The right-hand side of the equation is assumed to satisfy the conditions f(a)= f (a)= ··· =

x
f x (n) (a)=0.
n+1

ln x x d
Solution: y(x)= f(x).
n
2 n! x ln x dx
x
x + b
11. ln y(t) dt = f(x).
a t + b
This is a special case of equation 1.9.2 with g(x) = ln(x + b).

Solution: y(x)=(x + b)f (x)+ f (x).
xx x
x

12. ln(x/t) y(t) dt = f(x).
a
Solution:
2
x
2 d f(t) dt
y(x)= x .
πx dx a t ln(x/t)
x
y(t) dt

13. = f(x).
a ln(x/t)
Solution: x
1 d f(t) dt
y(x)= .
π dx a t ln(x/t)
© 1998 by CRC Press LLC
© 1998 by CRC Press LLC
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