Very recently, the system of differential equations governing the three-dimensional falling body problem (TDFBP) has been approximately solved. The previously obtained approximate solution was based on the fact that the Earth’s rotation (ER) is quite slow and hence all high order terms of $\omega$ in addition to the magnitude ${\omega }^{2}R$ were neglected, where $\omega$ is the angular velocity and $R$ is the radius of Earth. However, it is shown in this paper that the ignorance of such magnitudes leads, in many cases, to significant errors in the estimated falling time and other physical quantities. The current results are based on obtaining the exact solutions of the full TDFBP-system and performing several comparisons with the approximate ones in the relevant literature. The obtained results are of great interest and importance, especially for other planets in the Solar System or exterior planets, in which $\omega$ and/or ${\omega }^{2}R$ are of considerable amounts and hence cannot be ignored. Therefore, the present analysis is valid in analyzing the TDFBP near to the surface of any spherical celestial body.

This paper solves a generalized class of first-order fractional ordinary differential equations (1st-order FODEs) by means of Riemann–Liouville fractional derivative (RLFD). The principal incentive of this paper is to generalize some existing results in the literature. An effective approach is applied to solve non-homogeneous fractional differential systems containing $2n$ periodic terms. The exact solutions are determined explicitly in a straightforward manner. The solutions are expressed in terms of entire functions with fractional order arguments. Features of the current solutions are discussed and analyzed. In addition, the existing solutions in the literature are recovered as special cases of our results.

The existence of the advance parameter in a scalar differential equation prevents the application of the well-known standard methods used for solving classical ordinary differential equations. A simple procedure is introduced in this paper to remove the advance parameter from a special kind of first-order scalar differential equation. The suggested approach transforms the given first-order scalar differential equation to an equivalent second-order ordinary differential equation (ODE) without the advance parameter. Using this method, we are able to construct the exact solution of both the transformed model and the given original model. The exact solution is obtained in a wave form with specified amplitude and phase. Furthermore, several special cases are investigated at certain values/relationships of the involved parameters. It is shown that the exact solution in the absence of the advance parameter reduces to the corresponding solution in the literature. In addition, it is declared that the current model enjoys various kinds of solutions, such as constant solutions, polynomial solutions, and periodic solutions under certain constraints of the included parameters.

This paper solves a nonhomogeneous version of the pantograph equation. The nonhomogeneous term is taken as a polynomial of degree n with arbitrary coefficients. The nonhomogeneous pantograph equation is successfully converted to the standard homogeneous version by means of a simple transformation. An explicit formula is derived for the coefficients of the assumed transformation. Accordingly, the solution of the nonhomogeneous version is obtained in different forms in terms of power series, in addition to exponential functions. The obtained solution in power-series form is investigated to produce exact solutions for several examples under specific relationships between the involved parameters. In addition, exact solutions in terms of trigonometric and hyperbolic functions are determined at a certain value of the proportional delay parameter. The obtained results may be reported for the first time for the present nonhomogeneous version of the pantograph equation and can be further applied to include other versions with different nonhomogeneous terms.

Delay differential equations are fundamental tools to modeling various real-world problems. A particular type of these models is considered in this paper in the form y’(t) = ay(t) + aeaty(at), where a is a proportional delay parameter. Solving delay equations is usually a difficult task. This is because there are no standard/well-known methods for solving such kind of equations. This paper proposes a simple procedure to solve the above delay equation. The solution is obtained in closed form which is optimal. The suggested analysis can be invested to analyze more complex models in physics and engineering sciences.