What is the equation for natural frequencies of a rod in torsional oscillation?

[KMD]

Homework Statement

Determine the equation for the natural frequencies of a uniform rod in torsional oscillation with one end fixed and the other end free

Homework Equations

Theta = [A*sin(omega*(rho/G)^(1/2)*x)+B*cos((omega*(rho/G)^(1/2)*x)]*[C*sin(omega*t)+D*cos(omega*t)]

The Attempt at a Solution

This is an example problem in the book Theory of vibrations with applications. I don't understand why they stard with the equation
[A*sin(omega*(rho/G)^(1/2)*x)+B*cos((omega*(rho/G)^(1/2)*x)]*sin(omega*t)
instead of the equation containing the constants C and D.

 

[Asim Riaz]

Can someone explain why? I am assuming that the equation for the natural frequencies is omega=(rho/G)^(1/2)