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A First Course in Differential Equations

J. David Logan
Publication Date: 
Number of Pages: 
Undergraduate Texts in Mathematics
[Reviewed by
William J. Satzer
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Over the last several years it has become common to find introductory texts on ordinary differential equations that have ballooned to incorporate student projects, computer exercises, supplementary manuals, and guides to web content as well as large collections of techniques and applications. At the same time, typical undergraduate courses are still only one semester long and generally aim to introduce basic concepts, solution methods, some applications and a bit of the qualitative study of nonlinear equations. The discrepancy is considerable.

The author of the current book has chosen instead to make his book fit the intended course without adding a lot of frills. He aims to provide material for a one-semester course that emphasizes the basic ideas, solution methods, and an introduction to modeling. Throughout he tries to maintain a tight connection between mathematics and applications.

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. Students are gently introduced to differential equations largely in the context of modeling physical, biological and chemical systems. The author takes pains to describe the basic concepts slowly and carefully. He does as good a job as I’ve seen discussing existence and uniqueness results at this level. He raises the question, indicates why the issue can be important, and provides a couple of examples, then refers to more advanced treatments, and stops. The treatment of linear algebra is pretty much bare bones; it enters only in the guise of matrix methods for handling systems of differential equations.

Given the author’s intentions, it is a little surprising to see an entire chapter on Laplace transforms. This would be of value to engineering students, but could be omitted without loss of continuity. As it is, the book has more material than could reasonably be treated in one semester.

Though the succinctness and directness of the treatment are admirable in principle, some important topics are treated very quickly indeed. For example, resonance gets less than four pages, second-order linear equations with variable coefficients barely two, the Cauchy-Euler equation about three, and power series solutions about the same. These sections barely outline the topic and essentially include one example apiece. Throughout the book there are several areas where the treatment is so terse that some students would need help to unwind the contents.

There are an adequate number of exercises at a range of level of difficulty. Solutions are provided for even-numbered exercises in an appendix. This book is worth a careful look as a candidate text for the next differential equations course you teach.

Bill Satzer ( is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.

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.-