This book, together with the linked YouTube videos, reviews a first course on differential equations. The main purpose is to help students prepare for their university exams. Theory is summarized, and the solutions of typical exam questions are demonstrated in YouTube videos. Additional practice questions are given and their solutions are presented in the Appendix. The topics covered, which can be studied independently, include various first-order differential equations, second-order differential equations with constant coefficients, the Laplace transform, power series solutions, Cauchy-Euler equations, systems of linear first-order equations, nonlinear differential equations, and Fourier series.

This review book, used in conjuction with free online YouTube videos, is designed to help students prepare for exams, or for self-study. The topics covered here are most of the standard topics covered in a first course in differential equations.

The chapters and sections of this review book, organized by topics, can be read independently. Each chapter or section consists of three parts: (1) Theory; (2) YouTube Example; and (3) Additional Practice. In Theory, a summary of the topic and associated solution method is given. It is assumed that the student has seen the material before in lecture or in a standard textbook so that the presentation is concise. In YouTube Example, an online YouTube video illustrates how to solve an example problem given in the review book. Students are encouraged to view the video before proceeding to Additional Practice, which provides additional practice exercises similar to the YouTube example. The solutions to all of the practice exercises are given in this review book’s Appendix.

For students who self-study, or desire additional explanatory materials, a complete set of free lecture notes by the author entitled An Introduction to Differential Equations can be downloaded by clicking HERE. This set of lecture notes also contains links to additional YouTube tutorials. The lecture notes and tutorials have been extensively used by the author over several years when teaching an introductory differential equations course at the Hong Kong University of Science and Technology.

1. First-order differential equations
   1. Separable equations
   2. Linear equations
   3. Exact equations
   4. Bernoulli equations
   5. First-order homogeneous equations
   6. Riccati equations
2. Second-order differential equations with constant coefficients
   1. Homogeneous equations
   2. Inhomogeneous equations
3. The Laplace transform
4. Power series solutions
5. Cauchy-Euler equations
6. Systems of linear equations
7. Nonlinear differential equations
   1. Fixed points and linear stability analysis
   2. Bifurcation theory
8. Fourier series
9. Appendix A: Table of Laplace transforms
10. Appendix B: Answers to Additional Practice