Modern Differential Equations

Interestingly, this metamorphosis in the teaching of differential equations occurred relatively "overnight" and coincided with our professional careers at Georgia Southern University. Our interest in the use of technology in the mathematics classroom began in 1990 when we started to use computer laboratories and demonstrations in our calculus, differential equations, and applied mathematics courses. Over the past ten years we have learned some ways of how to and how not to use technology in the mathematics curriculum. In the early stages, we simply wanted to show students how they could solve more difficult problems by using a computer algebra system so that they could be exposed to the technology. However, we soon realized that we were missing the great opportunity of allowing students to discover aspects of the subject matter on their own. We revised our materials to include experimental problems and thought-provoking questions in which students are asked to make conjectures and investigate supporting evidence. We also developed application projects called Differential Equations at Work, not only to emphasize technology, but also to improve the problem-solving and communication skills of our students. To preserve the "wow" aspects of technology, we continue to use it to observe solutions in classroom demonstrations through such things as animating the motion of springs and pendulums. These demonstrations not only grab the attention of students, but also help them to make the connection between a formula and what it represents.

In presenting our findings to colleagues around the country, we quickly found out that others were interested in our work. As a result, we decided to develop a differential equations textbook to share this work with those who share our desire to improve mathematics education. This book is a culmination of years of "trials and tribulation" as the differential equations students at Georgia Southern can attest. Our hope is that its use will inspire students to open their eyes to the exciting discoveries that differential equations offer.

This book is designed to serve as a text for beginning courses in differential equations. Usually, introductory differential equations courses are taken by students who have successfully completed a first-year calculus course, and this text is written at a level readable for them.

Contents

• Introduction to differential equations
• First-order equations
• Applications of first-order equations
• High-order equations
• Applications of higher order equations
• Systems of differential equations
• Applications of first-order systems
• Laplace transforms
• Eigenvalue problems and Fourier series
• Partial differential equations
• Answers to selected exercises
• Index