Page 406 - Handbook Of Integral Equations

b

6.2-4. Equations of the Form y(x)+ G(···) dt = F (x)
a
b
30. y(x)+ f(t)y(t)y(xt) dt =0.
a
◦
1 . Solutions:
1 C (I 1 – I 0 )x + I 1 – I 2 C
y 1 (x)= – x , y 2 (x)= 2 x ,
I 0 I 0 I 2 – I
1
b

I m = f(t)t 2C+m dt, m =0, 1, 2,
a
where C is an arbitrary constant.
k
There are more complicated solutions of the form y(x)= x C n B k x , where C is an
k=0
arbitrary constant and the coefficients B k can be found from the corresponding system of
algebraic equations.

2 . A solution:
◦
β
(I 1 – I 0 )x + I 1 – I 2 C
y 3 (x)= x ,
I 0 I 2 – I 2
1
b
I m = f(t)t 2C+mβ dt, m =0, 1, 2,
a
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 )
y 4 (x)= ,
J 0 J 2 – J 2
1
b

m
J m = f(t)t 2C (ln t) dt, m =0, 1, 2,
a
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.
4 . The equation also has the trivial solution y(x) ≡ 0.
◦
β
5 . The substitution y(x)= x w(x) leads to an equation of the same form,
◦
b
2β
w(x)+ g(t)w(t)w(xt) dt =0, g(x)= f(x)x .
a

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 386