Page 397 - Handbook Of Integral Equations

◦
2 . A solution:
β
(I 1 – I 0 )x + I 1 – I 2 C A
y 3 (x)= x , I m = , m =0, 1, 2,
I 0 I 2 – I 2 2C + mβ +1
1
where C and β are arbitrary constants.
kβ
There are more complicated solutions of the form y(x)= x C n D k x , where C and β
k=0
are arbitrary constants and the coefficients D k can be found from the corresponding system
of algebraic equations.
◦
3 . A solution:
C
x (J 1 ln x – J 2 ) 1 2C m
y 4 (x)= , J m = t (ln t) dt, m =0, 1, 2,
J 0 J 2 – J 2
1 0
where C is an arbitrary constant.
k
There are more complicated solutions of the form y(x)= x C n E k (ln x) , where C is
k=0
an arbitrary constant and the coefficients E k can be found from the corresponding system of
algebraic equations.

∞

36. y(x)+ A y(t)y(xt) dt =0.
1
This is a special case of equation 6.2.30 with f(t)= A, a = 1, and b = ∞.

∞
β
37. y(x)+ λ y(t)y(xt) dt = Ax .
1
This is a special case of equation 6.2.31 with f(t)= λ, a = 0, and b =1.
1

38. y(x)+ A y(t)y(x + λt) dt =0.
0
This is a special case of equation 6.2.35 with f(t) ≡ A, a = 0, and b =1.
1 . A solution:
◦
C(λ +1) Cx
y(x)= e ,
A[1 – e C(λ+1) ]
where C is an arbitrary constant.

m
◦
2 . There are more complicated solutions of the form y(x)= e Cx n B m x , where C is an
m=0
arbitrary constant and the coefficients B m can be found from the corresponding system of
algebraic equations.
∞

39. y(x)+ A y(t)y(x + λt) dt =0, λ >0, 0 ≤ x < ∞.
0
This is a special case of equation 6.2.35 with f(t) ≡ A, a = 0, and b = ∞.
A solution:
C(λ +1) –Cx
y(x)= – e ,
A
where C is an arbitrary positive constant.

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 377