In this paper, we solve the Duffing equation for given initial conditions. We introduce the concept of the discriminant for the Duffing equation and we solve it in three cases depending on sign of the discriminant. We also show the way the Duffing equation is applied in soliton theory.

The Sunyaev–Zel’dovich (SZ) effect is a global distortion of the Cosmic Microwave Background (CMB) spectrum as a result of its interaction with a hot electron plasma in the intracluster medium of large structures gravitationally viralized such as galaxy clusters (GC). Furthermore, this hot gas of electrons emits X-rays due to its fall in the gravitational potential well of the GC. The analysis of SZ and X-ray data provides a method for calculating distances to GC at high redshifts. On the other hand, many galaxies and GC produce a Strong Gravitational Lens (SGL) effect, which has become a useful astrophysical tool for cosmology. We use these cosmological tests in addition to more traditional ones to constrain some alternative dark energy (DE) models, including the study of the history of cosmological expansion through the cosmographic parameters. Using Akaike and Bayesian Information Criterion, we find that the $wCDM$ and $\mathsf{\Lambda }CDM$ models are the most favoured by the observational data. In addition, we found at low redshift a peculiar behavior of slowdown of the universe, which occurs in dynamical DE models when we use data from GC.