Differential Equations: A First Course on ODE and a Brief Introduction to PDE

This book is meant to serve as a textbook for a basic introductory course in differential equations. It has many non-standard topics, which makes it quite unique within the sea of differential equations textbooks. The classical topics are covered (e.g. first order equations, exact equations, second order equations, etc.), but they are covered with more theoretical detail than is expected for a “typical” differential equations textbook. For instance, Section 3.4 is an Appendix containing detailed proofs of the classical existence and uniqueness theorems. The proofs in the book are generally provided with sufficient detail to discuss in class if an instructor desires. Additionally, each Chapter contains a very nice selection of exercises with varying difficulty.

 

There is quite a bit of material covered within the book. This means, though, that classical topics are covered in slightly less detail than one typically expects in a differential equations textbook. For example, the Chapter on Laplace transforms is unusually short (on purpose).  

 

The authors say there is also enough material for a one semester graduate course in differential equations. The material in the book for such a course includes Sturm-Liouville theory, oscillation theory, an introduction to partial differential equations and calculus of variations. The inclusion of this last topic in a differential equations textbook is particularly interesting, and is presented at a basic level which should be accessible to undergraduate students. The chapters on partial differential equations include the three basic equations: the Laplace equation, the heat equation, and the wave equation. Another interesting topic that is included is the maximum principle for the Laplace equation; this was quite a surprise!

 

The authors assume knowledge of calculus and some linear algebra, but also include some relevant background material in the first chapter. The book is written to be flexible to the needs of the instructor. For instance, instructors are able to focus more on qualitative aspects of differential equations if desired. Overall, this book is focused more on theoretical aspects of differential equations, but again can be adapted by the instructor as needed. Advanced topics are included, but are presented at a fairly elementary level.

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Eric Stachura is currently an Assistant Professor of Mathematics at Kennesaw State

University. He is generally interested in analysis and partial differential equations.