Page 393 - Handbook Of Integral Equations

b

6.1-2. Equations of the Form G(···) dt = F (x)
a
1
12. y(t)y(xt) dt = A, 0 ≤ x ≤ 1.
0
This is a special case of equation 6.2.2 with f(t)=1, a = 0, and b =1.

◦
1 . Solutions:
√ √
y 1 (x)= A, y 2 (x)= – A,
√ √
y 3 (x)= A (3x – 2), y 4 (x)= – A (3x – 2),
√ √
2
2
y 5 (x)= A (10x – 12x + 3), y 6 (x)= – A (10x – 12x + 3).
◦
2 . The integral equation has some other solutions; for example,
√ √
A C A C
y 7 (x)= (2C +1)x – C – 1 , y 8 (x)= – (2C +1)x – C – 1 ,
C C
√ √
y 9 (x)= A (ln x + 1), y 10 (x)= – A (ln x + 1),
where C is an arbitrary constant.
3 . See 6.2.2 for some other solutions.
◦
1

β
13. y(t)y(xt ) dt = A, β >0.
0
◦
1 . Solutions:
√ √
y 1 (x)= A, y 2 (x)= – A,
√
√
y 3 (x)= B (β +2)x – β – 1 , y 4 (x)= – B (β +2)x – β – 1 ,

2A
where B = .
β(β +1)
2 . The integral equation has some other (more complicated solutions) of the polynomial
◦
k
form y(x)= n B k x , where the constants B k can be found from the corresponding system
k=0
of algebraic equations.
∞

–λ
14. y(t)y(xt) dt = Ax , λ >0, 1 ≤ x < ∞.
1
This is a special case of equation 6.2.3 with f(t)=1, a = 1, and b = ∞.
◦
1 . Solutions:
–λ
–λ
1
y 1 (x)= Bx , y 2 (x)= –Bx , λ > ;
2
–λ –λ 3
y 3 (x)= B (2λ – 3)x – 2λ +2 x , y 4 (x)= –B (2λ – 3)x – 2λ +2 x , λ > ;
2
√
where B = A(2λ – 1).
2 . For sufficiently large λ, the integral equation has some other (more complicated) solutions
◦
k
of the polynomial form y(x)= n B k x , where the constants B k can be found from the
k=0
corresponding system of algebraic equations. See 6.2.2 for some other solutions.
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 373