Page 32 - Handbook Of Integral Equations

x
√

33. 1+ b x – t y(t) dt = f(x).
a
Differentiating with respect to x, we arrive at Abel’s equation of the second kind 2.1.46:
b x y(t) dt

y(x)+ √ = f (x).
x
2 a x – t
x

√ √
34. t x – x t y(t) dt = f(x).
a
√
This is a special case of equation 1.9.11 with g(x)= x and h(x)= x.
x
√ √
35. At x + Bx t y(t) dt = f(x).
a
√
This is a special case of equation 1.9.12 with g(x)= x and h(t)= t.
x
y(t) dt
36. √ = f(x).
a x – t
Abel’s equation.
Solution:

1 d x f(t) dt f(a) 1 x f (t) dt

t
y(x)= √ = √ + √ .
π dx a x – t π x – a π a x – t
•
Reference: E. T. Whittacker and G. N. Watson (1958).
x 1
37. b + √ y(t) dt = f(x).
a x – t
Let us rewrite the equation in the form

x x
y(t) dt

√ = f(x) – b y(t) dt.
a x – t a
Assuming the right-hand side to be known, we solve this equation as Abel’s equation 1.1.36.
After some manipulations, we arrive at Abel’s equation of the second kind 2.1.46:

x x

b y(t) dt 1 d f(t) dt
y(x)+ √ = F(x), where F(x)= √ .
π x – t π dx x – t
a a
x
1 1

38. √ – √ y(t) dt = f(x).
a x t
1
This is a special case of equation 1.1.44 with µ = – .
2

Solution: y(x)= –2 x 3/2 x x a >0.
f (x) ,
x
A B
39. √ + √ y(t) dt = f(x).
a x t
1
This is a special case of equation 1.1.45 with µ = – .
2
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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