Page 407 - Handbook Of Integral Equations

b
β
31. y(x)+ f(t)y(t)y(xt) dt = Ax .
a
1 . Solutions:
◦
β
β
y 1 (x)= k 1 x , y 2 (x)= k 2 x ,
where k 1 and k 2 are the roots of the quadratic equation
b
2
Ik + k – A =0, I = f(t)t 2β dt.
a
◦
2 . Solutions:
β
y(x)= x (λx + µ),
where λ and µ are determined from the following system of two algebraic equations (this
system can be reduced to a quadratic equation):

2
I 2 λ + I 1 µ +1 = 0, I 1 λµ + I 0 µ + µ – A =0
b

where I m = f(t)t 2β+m dt, m =0, 1, 2.
a
m
3 . There are more complicated solutions of the form y(x)= x β n B m x , where the B m
◦
m=0
can be found from the corresponding system of algebraic equations.
b

32. y(x)+ f(t)y(t)y(xt) dt = A ln x + B.
a
This equation has solutions of the form y(x)= p ln x + q, where the constants p and q can be
found from a system of two second-order algebraic equations.

∞ x

33. y(x)+ f(t)y(t)y dt =0.
t
0
◦
1 . A solution:
–1
∞

C
y(x)= –kx , k = f(t) dt ,
0
where C is an arbitrary constant.
◦
2 . The equation has the trivial solution y(x) ≡ 0.
β
3 . The substitution y(x)= x w(x) leads to an equation of the same form,
◦
∞

x
w(x)+ f(t)w(t)w dt =0.
0 t
∞ x

b
34. y(x)+ f(t)y y(t) dt = Ax .
t
0
Solutions:
b
b
y 1 (x)= λ 1 x , y 2 (x)= λ 2 x ,
where λ 1 and λ 2 are the roots of the quadratic equation
∞

2
Iλ + λ – A =0, I = f(t) dt.
0
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 387