Page 425 - Handbook Of Integral Equations

b

10. y(xt)f t, y(t) dt = A sin(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.

b

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

12. y(x + βt)f t, y(t) dt = Ax + B, β >0.
a
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 f(t, pt + q) dt – A =0, (βpt + q)f(t, pt + q) dt – B = 0. (2)
a a
Different solutions of system (2) generate different solutions (1) of the integral equation.
b
–λx
13. y(x + βt)f t, y(t) dt = Ae , β >0.
a
Solutions:
y(x)= k n e –λx ,
where k n are roots of the algebraic (or transcendental) equation
b

–λt –βλt
kF(k) – A =0, F(k)= f t, ke e dt.
a
b

14. y(x + βt)f t, y(t) dt = A cos λx, β >0.
a
This equation has solutions of the form y(x)= p sin λx + q cos λx, where p and q are some
constants.
b

15. y(x + βt)f t, y(t) dt = A sin λx, β >0.
a
This equation has solutions of the form y(x)= p sin λx + q cos λx, where p and q are some
constants.
b
–µx
16. y(x + βt)f t, y(t) dt = e (A cos λx + B sin λx), β >0.
a
This equation has solutions of the form y(x)= e –µx (p sin λx + q cos λx), where p and q are
some constants.
b

17. y(x – t)f t, y(t) dt = Ax + B.
a
This equation has solutions of the form y(x)= px + q, where p and q are some constants.

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 405