Differential equations are the foundation of mathematical modeling. Whether it’s the motion of planets, the spread of a virus, or the flow of electricity, these equations provide a means to connect what we know about a system to how it will evolve over time or space. This chapter serves as your starting point for understanding why differential equations matter and how they’re built.
We’ll begin with the essential question: “What exactly is a differential equation?” From there, we’ll break the concept into manageable pieces: learning the vocabulary and identifying key components such as terms and coefficients. These ideas form the language that the rest of the book will speak.
Every differential equation involves two variables: a dependent variable (the unknown function we’re solving for) and an independent variable (the variable it depends on).
A term is any part of a differential equation separated by plus or minus signs. Terms can include derivatives, the dependent variable, constants, or other functions of the independent variable.
