This study investigates two transitions in cosmology: radiation-matter and matter-dark energy. For each transition, a parameter $\chi$ is employed, representing the ratio of the two energy densities involved in the relevant transition. The focus on the second transition is motivated by the need to understand an accelerating universe. In examining potential cosmic acceleration due to dynamic dark energy, a dynamical equation of state for dark energy is considered in terms of the ratio $\chi$ and the deceleration parameter $q$. The resulting system of equations is analyzed by varying parameters to investigate their influence on the evolution of the universe. To achieve cosmic acceleration, the... More >