The theory of differential equations arises from the study of physical phenomena. This field has various applications in science and engineering. The study of qualitative properties for each mathematical model plays an important role, attracting the attention of both theoretical and applied researchers. Normally, the most significant qualitative property to be studied first is the existence and uniqueness of the solutions of each mathematical model. However, proving existence and uniqueness results for mathematical models where the source function has a singularity is a difficult problem and requires many different techniques. In this paper, we establish some new conditions suitable to achieve the unique solution criterion for ordinary first-order differential equations. To obtain the desired results, we have improved the methods that have been used to prove the results in the work of Krasnosel'skii and Krein (Krasnoselskii and Krein, 1956). In addition, we also provide an example to illustrate the theoretical results.