Geometry of Differential Equations

The geometry of differential equations involves visualizing solutions as shapes or curves in space, where each point represents a possible state of a system. Instead of just solving equations algebraically, it studies how these solutions behave and relate to each other by examining structures like direction fields and phase portraits. This geometric perspective helps us understand the stability, patterns, and long-term behavior of solutions, providing intuitive insights into complex dynamic systems, such as population growth, physics, or engineering, through visual and spatial reasoning.