Mathematics · Ch 18 — Differential Equations

Introduction

18.1

Introduction

The Idea of a Differential Equation

In earlier classes, given $f(x)$ you learned to find its derivative $f'(x)$ . In integral calculus you reversed this: given $g$ , find $f$ such that $f'(x) = g(x)$ . This reverse problem can be written as

\frac{dy}{dx} = g(x), \quad \text{where } y = f(x) \qquad(1)

An equation of the form (1) is called a differential equation — the defining feature is that it involves a derivative.

Why Differential Equations Matter

Differential equations arise wherever we study change. They describe motion and fields in Physics, reaction rates in Chemistry, population growth in Biology, investment growth in Economics, and evolution over time in Anthropology and Geology. This wide applicability makes their study essential in modern science.

What This Chapter Covers

The general and particular solutions of a differential equation
How to form a differential equation from a given relation
Methods to solve a first-order, first-degree differential equation
Applications of differential equations in various fields

Note

First-order, first-degree means the highest derivative present is the first derivative, and that derivative appears only to the first power (no squares, cubes, or other exponents).

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