Page 424 - Handbook Of Integral Equations

b

5. y(xt)f t, y(t) dt = Ax + B.
a
1 . A solution:
◦
y(x)= px + q, (1)
where p and q are roots of the following system of algebraic (or transcendental) equations:

b b
p tf(t, pt + q) dt – A =0, q f(t, pt + q) dt – B = 0. (2)
a a
Different solutions of system (2) generate different solutions (1) of the integral equation.

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 (or transcendental) equations.
b

β
6. y(xt)f t, y(t) dt = Ax .
a
A solution:
β
y(x)= kx , (1)
where k is a root of the algebraic (or transcendental) equation

b

β

kF(k) – A =0, F(k)= t f t, kt β dt. (2)
a
Each root of equation (2) generates a solution of the integral equation which has the form (1).
b

7. y(xt)f t, y(t) dt = A ln x + B.
a
A solution:
y(x)= p ln x + q, (1)
where p and q are roots of the following system of algebraic (or transcendental) equations:

b b
p f(t, p ln t + q) dt – A =0, (p ln t + q)f(t, p ln t + q) dt – B = 0. (2)
a a
Different solutions of system (2) generate different solutions (1) of the integral equation.
b

β
8. y(xt)f t, y(t) dt = Ax ln x.
a
β
β
This equation has solutions of the form y(x)= px ln x + qx , where p and q are some
constants.
b

9. y(xt)f t, y(t) dt = A cos(β ln x).
a
This equation has solutions of the form y(x)= p cos(β ln x)+ q sin(β ln x), where p and q are
some constants.

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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