Differential equations are mathematical equations that relate a function to its derivatives, forming the backbone of mathematical modeling in physics, engineering, biology, and economics. They describe how quantities change continuously in response to various factors—whether modeling population growth, heat distribution, fluid dynamics, or electrical circuits. The key distinction lies between ordinary differential equations (ODEs), involving derivatives with respect to a single variable, and partial differential equations (PDEs), involving multiple variables. Understanding solution techniques—from separation of variables to numerical methods—enables practitioners to predict system behavior, analyze stability, and solve real-world problems that would be intractable through algebraic methods alone.