Page 124 - Handbook Of Integral Equations

1.9-3. Other Equations

x

n
37. g(x) – g(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

1 1 d
Solution: y(x)= g (x) f(x).

x
n! g (x) dx

x
x

38. g(x) – g(t) y(t) dt = f(x), f(a)=0.
a
Solution:
2 x

2 1 d f(t)g (t) dt
t

y(x)= g (x) √ .
x
π g (x) dx a g(x) – g(t)

x
x y(t) dt

39. √ = f(x), g >0.
x
a g(x) – g(t)
Solution:
1 d x f(t)g (t) dt

t
y(x)= √ .
π dx a g(x) – g(t)
x
e y(t) dt
λ(x–t)

40. √ = f(x), g >0.
x
a g(x) – g(t)
Solution:

1 λx d x e –λt f(t)g (t)
t
y(x)= e √ dt.
π dx a g(x) – g(t)
x

λ
41. [g(x) – g(t)] y(t) dt = f(x), f(a)=0, 0< λ <1.
a
Solution:
2
x
1 d g (t)f(t) dt sin(πλ)

t

y(x)= kg (x) , k = .
x
g (x) dx [g(x) – g(t)] λ πλ

x a
x
h(t)y(t) dt

42. = f(x), g >0, 0< λ <1.
x
a [g(x) – g(t)] λ
Solution:
sin(πλ) d x f(t)g (t) dt

t
y(x)= .
πh(x) dx a [g(x) – g(t)] 1–λ
x
t
µ
λ
43. K y(t) dt = Ax + Bx .
x
0
Solution:
A λ–1 B µ–1 1 λ–1 1 µ–1
y(x)= x + x , I λ = K(z)z dz, I µ = K(z)z dz.
I λ I µ 0 0
© 1998 by CRC Press LLC
© 1998 by CRC Press LLC
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