The synchronization is the adaptation in the motion of the self-sustained oscillators, which is caused by the weak interaction between the components of the system. This feature has been found in many natural and biological systems and has led scientists to describe how it works. Despite the many models presented to describe synchrony, the Kuramoto model is a simple mathematical yet practical model derived from the generalization of the relationships of two coupled oscillators. Although the first-order Kuramoto model is a good approach for describing synchronization in many natural phenomena, However, providing a model with phase matching and oscillator frequency simultaneously as well as inertia in the system helped to describe some natural models better and provide a good generalization for the Kuramoto model. This model is, in fact, an extension of the first-order Kuramoto model, which we examine here dynamically. Another important issue that is not considered because of the simplification of the problem in the Kuramoto model is the time lag between system components, which we know from experiments that this parameter can somehow influence the synchronization of the components. Here, we have attempted to investigate the performance of the system by using a second-order Kuramoto model and applying a time delay for two coupled oscillators and to compare the number of complete synchronous responses in both the first-order and second-order models in the presence of a time delay.