© 2011

A First Course in Differential Equations


  • New edition extensively revised and updated, making the book more accessible to sophomores and instructors

  • Includes a small amount of additional, elementary material put in the chapter on systems

  • Uses integrating factors to solve linear first order equations instead of by variation of parameters

  • Includes new applications, especially in biology, and 25 more pages of solutions to exercises

  • Contains MATLAB and Maple summaries


Part of the Undergraduate Texts in Mathematics book series (UTM)

Table of contents

  1. Front Matter
    Pages i-xviii
  2. J. David Logan
    Pages 1-72
  3. J. David Logan
    Pages 103-160
  4. J. David Logan
    Pages 161-191
  5. J. David Logan
    Pages 193-249
  6. J. David Logan
    Pages 251-297
  7. J. David Logan
    Pages 299-330
  8. Back Matter
    Pages 331-386

About this book


This concise and up-to-date textbook is designed for the standard sophomore course in differential equations. It treats the basic ideas, models, and solution methods in a user friendly format that is accessible to engineers, scientists, economists, and mathematics majors. It emphasizes analytical, graphical, and numerical techniques, and it provides the tools needed by students to continue to the next level in applying the methods to more advanced problems. There is a strong connection to applications with motivations in mechanics and heat transfer, circuits, biology, economics, chemical reactors, and other areas. Exceeding the first edition by over one hundred pages, this new edition has a large increase in the number of worked examples and practice exercises, and it continues to provide templates for MATLAB and Maple commands and codes that are useful in differential equations. Sample examination questions are included for students and instructors. Solutions of many of the exercises are contained in an appendix. Moreover, the text contains a new, elementary chapter on systems of differential equations, both linear and nonlinear, that introduces key ideas without matrix analysis. Two subsequent chapters treat systems in a more formal way. Briefly, the topics include: * First-order equations: separable, linear, autonomous, and bifurcation phenomena; * Second-order linear homogeneous and non-homogeneous equations; * Laplace transforms; and * Linear and nonlinear systems, and phase plane properties.


Laplace transforms Mathematical modelling Nonlinear systems Ordinary differential equations Phase plane phenomena Second-order differential equations Two-dimensional linear systems

Authors and affiliations

  1. 1.Department of MathematicsUniversity of Nebraska--LincolnLincolnUSA

About the authors

J. David Logan is Willa Cather Professor of Mathematics at the University of Nebraska Lincoln. His extensive research is in the areas of theoretical ecology, hydrogeology, combustion, mathematical physics, and partial differential equations. He is the author of six textbooks on applied mathematics and its applications, including Applied Partial Differential Equations, 2nd edition (Springer 2004) and Transport Modeling in Hydrogeochemical Systems (Springer 2001).

Bibliographic information


From the reviews:

"Logan has produced a well-crafted text, densely packed with interesting applications from diverse fields. The chapters cover (ordinary) differential equations, analytical solutions and approximations, second-order differential equations, Laplace transforms, linear and nonlinear systems. The material is well presented and introduces new concepts … . The text will certainly provide a good mental workout." (Christopher Howls, The Times Higher Education Supplement, November, 2006)

"This is a textbook for those who … want to learn some methods and techniques to handle mathematical models described by ordinary differential equations. … the book contains topics which are not included in other similar texts. … In addition, four appendices are added to complete the presentation … . The book is written in a pleasant and friendly style. It provides the reader with enough knowledge to engage with more advanced topics of differential equations … ." (Gheorghe Morosanu, Zentralblatt MATH, Vol. 1088 (14), 2006)

From the reviews of the second edition:

“Designed for standard second-year courses in differential equations, this text covers the basic ideas, models and solution methods in a format intended to be accessible to engineering, economics and mathematics students. Logan emphasises analytical, graphical and numerical techniques, and provides a strong connection to applications with motivations in mechanics and heat transfer, circuits, biology, economics and chemical reactors.” (Times Higher Education, May, 2011)

“The new edition covers essentially the same material as the first, with minor rearrangements, and it is about one-third longer. The coverage of linear systems in the plane is nicely detailed and illustrated. … Simple numerical methods are illustrated and the use of Maple and MATLAB is encouraged. There are over thirty pages of solutions and hints to selected exercises as well. … select Dave Logan’s new and improved text for my course.” (Robert E. O’Malley, Jr., SIAM Review, Vol. 53 (2), 2011)

“Aims to provide material for a one-semester course that emphasizes the basic ideas, solution methods, and an introduction to modeling. … The book that results offers a concise introduction to the subject for students of mathematics, science and engineering who have completed the introductory calculus sequence. … There are an adequate number of exercises … . Solutions are provided for … exercises in an appendix. This book is worth a careful look as a candidate text for the next differential equations course you teach.” (William J. Satzer, The Mathematical Association of America, January, 2011)