Analytical and Numerical Approaches to Solving First-Order Differential Equations

Abstract

First-order differential equations are fundamental in modeling dynamic systems in physics, biology, chemistry, and engineering. This paper presents a comprehensive study of analytical and numerical techniques used to solve first-order ordinary differential equations (ODEs). We discuss standard solution methods including separable, linear, and exact equations, along with integrating factors. Furthermore, we investigate numerical approaches such as Euler’s method and the Runge-Kutta method for cases where analytical solutions are difficult or impossible to obtain. Several real-world applications are presented, such as population growth modeling, chemical reaction rates, and RC electrical circuits. The study emphasizes the importance of choosing the appropriate method based on the nature of the problem and highlights the role of computer-aided computation in modern analysis. This research serves as a bridge between theoretical development and practical implementation, offering insights valuable to researchers, educators, and practitioners alike.

Cite as:

R. Panchal, & Awasare Anant. (2025). Analytical and Numerical Approaches to Solving First-Order Differential Equations. Journal of Applied Mathematics and Statistical Analysis, 6(2), 14–19. 

https://doi.org/10.5281/zenodo.16411181