For introductory courses in Differential Equations.
This best-selling text by these well-known authors blends the traditional algebra problem solving skills with the conceptual development and geometric visualisation of a modern differential equations course that is essential to science and engineering students. It reflects the new qualitative approach that is altering the learning of elementary differential equations, including the wide availability of scientific computing environments like Maple, Mathematica, and MATLAB. Its focus balances the traditional manual methods with the new computer-based methods that illuminate qualitative phenomena and make accessible a wider range of more realistic applications. Seldom-used topics have been trimmed and new topics added: it starts and ends with discussions of mathematical modeling of real-world phenomena, evident in figures, examples, problems, and applications throughout the text.
This 5th edition is a comprehensive and wide-ranging revision. In addition to fine-tuning of the exposition (both text and graphics) in numerous sections of Chapters 1 through 7, new applications have been inserted (including biological), and we have exploited throughout the new interactive computer technology that is now available to students on devices ranging from desktop and laptop computers to smart phones and graphing calculators, using computer algebra systems such as Mathematica, Maple, and MATLAB as well as online web sites such as Wolfram|Alpha.
However, with a single exception of a new section inserted in Chapter 5 (noted below), the class-tested table of contents of the book remains unchanged. Therefore, instructors' notes and syllabi will not require revision to continue teaching with this new edition. Chapters 8 through 10 are largely unchanged, apart from local refinements and insertion of some new graphics and an occasional new example.
A conspicuous feature of this edition is the insertion of about 80 new computer-generated figures, many of them illustrating how interactive computer applications with slider bars or touchpad controls can be used to change initial values or parameters in a differential equation, allowing the user to immediately see in real time the resulting changes in the structure of its solutions.
New Graphics
New figures inserted throughout illustrate the facility offered by modern computing technology platforms for the user to interactively vary initial conditions and other parameters in real time. Thus, using a mouse or touchpad, the initial point for an initial value problem can be dragged to a new location, and the corresponding solution curve is automatically redrawn and dragged along with its initial point. For instance, see the application modules for Sections 1.3 and 3.1. Using slider bars in an interactive graphic, the coefficients or other parameters in a linear system can be varied, and the corresponding changes in its direction field and phase plane portrait are automatically shown; for instance, see the Section 5.3 application module. The number of terms used from an infinite series solution of a differential equation can be varied, and the resulting graphical change in the corresponding approximate solution is shown immediately; for instance, see the Section 8.2 application module.
New Exposition
In a number of sections, new text and graphics have been inserted to enhance student understanding of the subject matter. For instance, see the treatments of separable equations in Section 1.4, linear equations in Section 1.5, isolated critical points in Sections 6.1 and 6.2, and the new example in Section 9.5 showing a vibrating string with a momentary "flat spot." Examples and accompanying graphics have been updated in Sections 2.4, 2.5, 2.6, 4.2, and 4.3 to illustrate new graphing calculators.
New Content
The single entirely new section for this edition is Section 5.3, which is devoted to the construction of a "gallery" of phase plane portraits illustrating all the possible geometric behaviors of solutions of the 2-dimensional linear system x' = Ax. In motivation and preparation for the detailed study of eigenvalue-eigenvector methods in subsequent sections of Chapter 5 (which then follow in the same order as in the previous edition), Section 5.3 shows how the particular arrangements of eigenvalues and eigenvectors of the coefficient matrix A correspond to identifiable patterns--"fingerprints," so to speak--in the phase plane portrait of the system x' = Ax. The resulting gallery is shown in the two facing pages of phase plane portraits shown in Figure 5.3.16 at the end of the section. The new 5.3 application module (on dynamic phase plane portraits) shows how students can use interactive computer systems to "bring to life" this gallery, by allowing initial conditions, eigenvalues, and even eigenvectors to vary in real time. This dynamic approach is then illustrated with several new graphics inserted in the remainder of Chapter 5.
Finally, for a new biological application, see the application module for Section 6.4, which now includes a substantial investigation of the nonlinear FitzHugh-Nagumo equations in neuroscience, which were introduced to model the behavior of neurons in the nervous system.