This book is designed to introduce readers to the fundamental concepts, techniques, and applications of differential equations, providing a solid foundation for further study or practical problem-solving. It is specifically intended for students of technical universities. The textbook is divided into five main chapters. In the first chapter, we will explore basic concepts and notation, establishing the language and framework necessary for understanding differential equations. Definitions and essential terminology will be clarified, accompanied by "Do it yourself" tasks to reinforce learning and ensure active engagement with the material. The subsequent chapters study specific classes of first order differential equations. Chapter 2 focuses on first order equations, including their definitions, key theorems, and special cases such as separable equations. Practical exercises are included to develop problem-solving skills. Chapter 3 covers linear differential equations, highlighting important methods such as the Euler-Lagrange approach and integrating factors. We will also examine Bernoulli equations, providing tools for solving a broad class of problems. In Chapter 4, we explore exact differential equations, discussing their properties and methods of solution. Techniques like integrating factors for nonexact equations are presented to expand your toolkit for tackling complex problems. The final chapter demonstrates the application of the theory to real-world modelling situations. We investigate how differential equations can describe phenomena in population dynamics, cooling processes, mixing, and elementary mechanics. These examples demonstrate the practical relevance of the concepts covered. An appendix on the review of integration techniques supports the main content, offering definitions, properties, tables of common integrals, and illustrative examples to strengthen your understanding.
This textbook aims to make the study of differential equations accessible, engaging, and applicable to various fields.
FOREWORD
CHAPTER 1. INTRODUCTION
1.1. Overview of differential equations
1.2. General definitions
1.3. Do it yourself tasks
CHAPTER 2. FIRST ORDER DIFFERENTIAL EQUATIONS
2.1. Basic definitions and theorems
2.2. Equation with separable variables
2.3. First order non-separable equations – solutions by substitutions
2.4. Do it yourself tasks
CHAPTER 3. LINEAR FIRST ORDER DIFFERENTIAL EQUATIONS
3.1. Basic terminology
3.2. Euler-Lagrange method for solving the linear first order differential equation
3.3. Integrating factor for linear first order equations
3.4. Bernoulli differential equation
3.5. Do it yourself tasks
CHAPTER 4. EXACT DIFFERENTIAL EQUATION
4.1. Basic concepts and definitions
4.2. Method of solution to the exact differential equation
4.3. Integrating factor for nonexact equation
4.4. Do it yourself tasks
CHAPTER 5. MODELING USING FIRST ORDER DIFFERENTIAL EQUATIONS
5.1. Population problems
5.2. Cooling problems
5.3. Mixing problems
5.4. Elementary mechanics problems
6. APPENDIX. REVIEW OF INTEGRATION
6.1. Basic definitions and integral rules
6.2. Table of common integrals
6.3. Examples
ANSWERS FOR SELECTED TASKS
BIBLIOGRAPHY
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