Stochastic differential equation of a falling body

Question

It's well known the motion of a falling body in a constant gravity model, for high speed is given by: $$m\ddot{x}(t)=g-\beta\dot{x}(t)^2$$ where $\beta$ is he drag coefficient. In a turbolent flow we can assume the drag coefficient $\beta$ is a gaussian distribuited random variable around a mean value $\beta_0$. So the previous equation becomes: $$m\ddot{x}(t)=g-\beta(t)\dot{x}(t)^2$$ where $\beta(t)=N(\beta_0,\sigma)$ How can the last equation be solved assuming thi behaviour of $\beta(t)$? Thanks in advance.