A First Course in Differential Equations

'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. 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.

List of contents

Preface to the Second Edition.- To the Student.- 1. Differential Equations and Models.- 1.1 Introduction.- 1.2 General Terminology.- 1.2.1 Geometrical Interpretation.- 1.3 Pure Time Equations.- 1.4 Mathematical Models.- 1.4.1 Particle Dynamics.- 1.5 Separation of Variables.- 1.6 Autonomous Differential Equations.- 1.7 Stability and Bifurcation 1.8 Reactors and Circuits.- 1.8.1 Chemical Reactors.- 1.8.2 Electrical Circuits 2. Linear Equations and Approximations.- 2.1 First-Order Linear Equations.- 2.2 Approximation of Solutions.- 2.2.1 Picard Iteration .- 2.2.2 Numerical Methods.- 2.2.3 Error Analysis.- 3. Second-Order Differential Equations.- 3.1 Particle Mechanics 3.2 Linear Equations with Constant Coefficients.- 3.3 The Nonhomogeneous Equation
3.3.1 Undetermined Coefficients.- 3.3.2 Resonance.- 3.4 Variable Coefficients.- 3.4.1 Cauchy-Euler Equation.- 3.4.2 Power Series Solutions .- 3.4.3 Reduction of Order .- 3.4.4 Variation of Parameters.- 3.5 Boundary Value Problems and Heat Flow .- 3.6 Higher-Order Equations.- 3.7 Summary and Review.- 4. Laplace Transforms.- 4.1 Definition and Basic Properties.- 4.2 Initial Value Problems.- 4.3 The Convolution Property.- 4.4 Discontinuous Sources.- 4.5 Point Sources.- 4.6 Table of Laplace Transforms.- 5. Systems of Differential Equations.- 5.1 Linear Systems.- 5.2 Nonlinear Models.- 5.3 Applications.- 5.3.1 The Lotka-Volterra Model.- 5.3.2 Models in Ecology.- 5.3.3 An Epidemic Model.- 5.4 Numerical Methods.- 6. Linear Systems.- 6.1 Linearization and Stability.- 6.2 Matrices .- 6.3 Two-Dimensional Linear Systems.- 6.3.1 Solutions and Linear Orbits.- 6.3.2 The Eigenvalue Problem.- 6.3.3 Real Unequal Eigenvalues.- 6.3.4 Complex Eigenvalues.- 6.3.5 Real, Repeated Eigenvalues.- 6.3.6 Stability.- 6.4 Nonhomogeneous Systems .- 6.5 Three-Dimensional Systems .- 7. Nonlinear Systems.- 7.1 Linearization Revisited.- 7.1.1 Malaria .- 7.2 Periodic Solutions.- 7.2.1 The Poincar e-Bendixson Theorem.- Appendix A. References.- Appendix B. Computer Algebra Systems.- B.1 Maple.- B.2 MATLAB.- Appendix C. Sample Examinations.- D. Index.-

About the author

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.

Report

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)

Product details

• Authors: David Logan, J. David Logan
• Publisher: Springer, Berlin
• Languages: English
• Product format: Paperback / Softback
• Released: 18.12.2012
• EAN: 9781461427223
• ISBN: 978-1-4614-2722-3
• No. of pages: 386
• Dimensions: 155 mm x 236 mm x 22 mm
• Illustrations: 1 Tabellen
• Series: Undergraduate Texts in Mathematics; Undergraduate Texts in Mathematics
• Subject: Natural sciences, medicine, IT, technology > Mathematics > Analysis