Abstract
Differential equations are an essential mathematical tool for modeling and analyzing phenomena across various scientific and technological disciplines. This article presents a comprehensive review of the foundations of ordinary (ODEs) and partial differential equations (PDEs), addressing their analytical, numerical, and computational solution methods. Their applications in classical and quantum physics, mathematical biology, economics, engineering, and computational sciences are explored, emphasizing their role in fluid dynamics, neuronal system modeling, and financial markets. Furthermore, their connection to nonlinear dynamical systems and chaos theory is analyzed, highlighting key concepts such as sensitivity to initial conditions, strange attractors, and bifurcations. Finally, future perspectives in differential equation research are discussed, including the impact of artificial intelligence, deep learning, and geometric methods in solving complex problems.