Preface Introduction
📝: 🎧 Listen.
Differential equations sit at the heart of how we describe change. Whether we’re tracking the spread of a disease, predicting the motion of a planet, or modeling the charge in an electrical circuit, differential equations provide the language and the structure to make sense of how systems evolve. They connect the algebra you already know with the calculus you’ve learned, and they open the door to ideas that reach across science, engineering, and beyond.
This book is designed to guide you through that world step by step. We begin with the basics—what a differential equation is, how to classify it, and what it means to solve one. From there, we explore the core solution methods for first-order equations, building intuition before moving into more advanced topics such as higher-order equations, Laplace transforms, and systems of equations. Along the way, you’ll see not only how to solve different types of problems, but also why the methods work and when each tool is the right one to use.
But not every equation has a neat solution you can write down. That’s why this book also covers qualitative methods for understanding solution behavior and numerical methods for approximating solutions when analytic solutions are unavailable. These approaches shift the focus to the broader picture: what the solutions mean, how they behave, and what they reveal about the systems they describe.
Above all, this book aims to stay approachable and conversational. You’ll find worked examples, visual tools, and quick-reference summaries to help you along the way. Whether you are here as a mathematician, an engineer, a scientist, or simply someone curious about how mathematics explains the world, this book will help you not just solve differential equations, but truly understand them.
