Differential Equations | Class 12 Mathematics Notes

Introduction

Differential equations are mathematical equations that involve an unknown function and its derivatives. They establish a relationship between a function and its rates of change, making them fundamental tools in modeling various natural and physical phenomena. The unknown function typically represents a physical quantity that varies with respect to one or more independent variables, and the derivatives represent the rates at which this quantity changes. For example, in physics, differential equations describe motion, heat flow, and wave propagation; in biology, they model population growth and decay; in economics, they represent changing market conditions. The study of differential equations enables us to find these unknown functions, which satisfy the given relationships involving their derivatives. This chapter focuses primarily on first-order differential equations, which involve the first derivative of the unknown function. Understanding these equations is crucial as they form the foundation for more complex differential equations encountered in higher studies.

📊 Diagram: No specific diagrams in this introductory section, but conceptual illustrations often show a curve representing an unknown function y = f(x) and its slope dy/dx at various points.

🧪 Activity: No specific activity in this section.

🔗 Connection: This introduction sets the stage for understanding the basic concepts of differential equations, including order, degree, and types of solutions, which are discussed in the next section.