Non-Gaussian noise widely exists in many kinds of nonlinear systems. The study about the non-stationary state evolution behavior of the system driven by non-Gaussian noise can help us to understand its inherent evolution mechanism more deeply. In this paper, we investigate the non-stationary state evolution problem of the non-linear dynamical system driven by both non-Gaussian noise and Gaussian white noise. First, the non-linear dynamical system is linearized in the initial area by using the $\Omega$-expansion of the Green function. Then, we obtain the expression for the approximate non-stationary state solution through the eigenvalue and eigenvector theory. Finally, taking the Logistic model as an example, we examine the influences of the non-Gaussian noise intensity, the correlation time and the deviation parameter on the non-stationary state solution and its mean. The results show that when the Logistic model is used to describe the growth of product output, the non-stationary state solution can better reflect the evolution behavior of the product output near the unstable point.