Page 434 - Handbook Of Integral Equations

b

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

b

p + p tf(t, pt + q) dt – A =0,
a
b (2)

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

b
β
47. y(x)+ 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

k + 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

48. y(x)+ 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

p + p f(t, p ln t + q) dt – A =0,
a
b (2)

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

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

β
β
β
p + p t f(t, pt ln t + qt ) dt = A,
a
b (2)

β
β
β
β
q + (pt ln t + qt )f(t, pt ln t + qt ) dt =0.
a
Different solutions of system (2) generate different solutions (1) of the integral equation.
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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