In an approach to construct coherent states proposed recently, there is the requirement to solve a time-dependent ordinary differential equation (TDODE). Based on the consideration that the equation possesses SU(1,1) dynamical symmetry, we propose a uniform method for solving the TDODE exactly by means of the theory of algebraic dynamics, whatever the time-dependent frequency takes what form as a function of time. Then, after impacting the time-dependent frequency take a specific form, which is a sudden change function, we obtain the exact solution with the method. Next, we construct the corresponding coherent states with the help of this exact solution. Then the evolution of quantum fluctuation (quantum noise) is discussed. According to the property of the curve of the uncertainty of momentum operator, we point out some properties which can be utilized when preparing the squeezed states of this quntum system. At last, the evolution of the uncertainty relation with time is discussed.