First-Order Ordinary Differential Equations: Comprehensive Theory, Methods, and Modern Applications

This extensive review provides a thorough examination of first-order ordinary differential equations (ODEs), covering fundamental theoretical concepts, diverse analytical solution techniques, stability analysis methods, numerical approximation algorithms, and interdisciplinary applications. The paper systematically explores classical models including exponential growth, logistic dynamics, and cooling laws, while extending to advanced topics such as bifurcation analysis, stochastic exten- sions, and modern computational approaches. Special attention is given to the in- terplay between analytical and numerical methods, with practical examples drawn from ecology, physics, engineering, and biomedical sciences. The work serves as both an educational resource for students and a reference for researchers and prac- titioners working with dynamical systems across scientific domains.