# Preface

No doubt, the topic of differential equations has become the most widely used mathematical tool in modeling of real world phenomenon. Therefore, the corresponding course has been taught by universities around the world for over two hundred years, typically, as a two-semester course. Since this tutorial is for second semester course, it is assumed that a user is familiar with material and basic terminology from the first part. To make exposition friendly, the text contains many links to other web sites such as Wikipedia, Wolfram MathWorld, Wikiversity, Encyclopedia of Mathematics, and other sources.

This tutorial, accompanied by the textbook Applied Differential Equations. The Primary Course by Vladimir Dobrushkin, CRC Press, 2015; http://www.crcpress.com/product/isbn/9781439851043, corresponds to the second course, which at Brown University is APMA 0340 (Applied Mathematics -- II). This tutorial is primarily for students who have some experience using Mathematica. If you have never used Mathematica before and would like to learn more of the basics for this computer algebra system, it is strongly recommended looking at the APMA 0330 tutorial.

The objectives of this tutorial are:

• To acquire the reader with the differential equations topic. Since the focus is made on various applications, this tutorial does not usually contain detailed proofs and instead refers the user to other sources.
• To provide information about most important tools and techniques used in ordinary and partial differential equations beyond which published textbooks typically cover. Therefore, the This web page could be used as a reference to the differential equations and their applications.
• To assist the user in learning a particular software along with studying the differential equations.

As any software, Mathematica keeps in its cache all input variables and outputs. To reuse them, the user must either to redefine all variables or clear them with either the dedicated command Clear or quit the kernel using Evaluation within Control Panel. The commands in this tutorial are written in bold black font, while Mathematica output is in regular fonts. This means that you can copy and paste all commands into your Mathematica notebook, change the parameters, and run them. You are free to use the scripts to your needs for learning how to use the Mathematica programs, and have the right to distribute this tutorial and refer to this tutorial with appropriate credit. Any comments and/or contributions for this tutorial are welcome; you can send your remarks to <Vladimir_Dobrushkin@brown.edu>

This tutorial contains software programs that are free: you can redistribute codes and/or modify scripts under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. This tutorial is distributed in the hope that its material and codes will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. For full version of GNU General Public License, see https://www.gnu.org/licenses/gpl-3.0.en.html.

Introduction to Linear Algebra with Mathematica

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Richard H. Enns and George C. McGuire,
Nonlinear Physics with Mathematica for Scientists and Engineers, Birhauser, Boston, 2001.

Mark S. Gockenbach
Mathematica Tutorial to accompany the book
SIAM, 2010
http://www.math.mtu.edu/~msgocken/pdebook2/mathtut2.pdf

Alfred Gray, Mike Mezzino, and Mark Pinsky
Introduction to Ordinary Differential Equations with Mathematica: An Integrated Multimedia Approach
Springer
ISBN-10: 0387944818 | ISBN-13: 978-0387944814

Brian R. Hunt, Ronald L. Lipsman, John E. Osborn, Donald A. Outing, Jonathan Rosenberg
Differential Equations with Mathematica, 3rd Edition
ISBN: 978-0-471-77316-0

Leonid Shifrin