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.

Fractional differential equations are an important branch of mathematics and have been considered under many different fractional derivatives. Among them, differential equations with Riesz-Caputo fractional derivatives have also attracted the attention of many researchers. Studying differential equations that may have singularity coefficients is more difficult than usual because they require several complex techniques. In the present paper, we consider a nonlinear pantograph differential equation where the source function may have a temporal singularity. Using the contraction principle, we prove that the problem has a unique solution under some appropriate conditions. Furthermore, we define a new type of Ulam-Hyers stability and show the main equation of the problem is stable in the mentioned sense. To obtain the main results, a new inequality is proposed and proved. Some examples are constructed to confirm the validity and feasibility of the theoretical results.

In physics, the majority of natural events have been researched and described using differential equations, each having its own initial and boundary conditions. These differential equations contain a large number of fundamental constants as well as other model parameters. They add to the equation's complexity and rounding errors, making the problem more difficult to solve. In this work, we provide a method for transforming these physics differential equations into dimensionless equations, which are significantly simpler. Nondimensionalization, by suitably substituting variables, is the process of removing some or all of the physical dimensions from an equation that contains physical quantities. Some benefits of these dimensionless equations include that they are simpler to identify when using well-known mathematical methods, need less time to compute, and do not round off errors. Through several examples we discuss, this method is useful not just in quantum mechanics but also in classical physics.

In this paper we discus on a Lyapunov-type inequality for a fractional differential equation involving sequential generalized Caputo fractional derivatives with boundary conditions. The results presented in this paper is new to the corresponding results in the literature.