Differential Equations: Concepts and Applications

Definition and Purpose of Differential Equations

A differential equation is an equation that relates one or more unknown functions to their derivatives. The functions often represent physical quantities, while the derivatives describe their rates of change. By expressing relationships between quantities and how they change, differential equations provide a mathematical framework for describing dynamic systems and natural laws. Differential equations are used to model phenomena in fields such as physics, engineering, economics, and biology, including motion, heat flow, wave propagation, population change, and financial systems. Their study focuses on finding solutions and understanding solution properties, such as existence, uniqueness, stability, and long-term behavior. When exact formulas cannot be found, numerical methods can approximate solutions using computers.

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Unknown Functions and Their Derivatives

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