In a number of sections, we have added new text and graphics to enhance student understanding of the subject matter. For instance, see the new introductory treatments of separable equations in Section 1.4 (page 30), of linear equations in Section 1.5 (page 46), and of isolated critical points in Sections 9.1 (page 503) and 9.2 (page 514). Also we have updated the examples and accompanying graphics in Sections 2.4–2.6, 7.3, and 7.7 to illustrate modern calculator technology.

New figures throughout the text illustrate the capability that modern computing technology platforms offer to vary initial conditions and other parameters interactively. These figures are accompanied by detailed instructions that allow students to recreate the figures and make full use of the interactive features. For example, Section 7.4 includes the figure shown, a Mathematica-drawn phase plane diagram for a linear system of the form $\text{x}\text{'}=\text{Ax}$; after putting the accompanying code into Mathematica, the user can immediately see the effect of changing the initial condition by clicking and dragging the “locator point” initially set at (4,2).

Similarly, the application module for Section 5.1 now offers Matlab and TI-Nspirelinebreak graphics with interactive slider bars that vary the coefficients of a linear differential equation. The Section 11.2 application module features a new Matlab graphic in which the user can vary the order of a series solution of an initial value problem, again immediately displaying the resulting graphical change in the corresponding approximate solution.

The single entirely new section for this edition is Section 7.4, 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 $\text{x}\text{'}=\text{Ax}$. In motivation and preparation for the detailed study of eigenvalue-eigenvector methods in subsequent sections of Chapter 7 (which then follow in the same order as in the previous edition), Section 7.4 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. The resulting gallery is shown in the two pages of phase plane portraits in Figure 7.4.16 (pages 417–418) at the end of the section. The new 7.4 application module (on dynamic phase plane portraits, 421) 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 7.

Finally, for a new biological application, see the application module for Section 9.4, which now includes a substantial investigation (551) of the nonlinear FitzHugh–Nagumo equations of neuroscience, which were introduced to model the behavior of neurons in the nervous system.

Many of the examples and problems are now organized under headings that make the topic easy to see at a glance. This not only adds to the readability of the book, but it also makes it easier to choose in-class examples and homework problems. For instance, most of the text examples in Section 1.4 are now labelled by topic, and the same is true of the wealth of problems following this section.

The effectiveness of the application modules located throughout the text is greatly enhanced by the supplementary material found at the new Expanded Applications website. Nearly all of the application modules in the text are marked with and a unique “tiny URL”—a web address that leads directly to an Expanded Applications page containing a wealth of electronic resources supporting that module. Typical Expanded Applications materials include an enhanced and expanded PDF version of the text with further discussion or additional applications, together with computer files in a variety of platforms, including Mathematica, Maple, Matlab, and in some cases Python and/or TI calculator. These files provide all code appearing in the text as well as equivalent versions in other platforms, allowing students to immediately use the material in the Application Module on the computing platform of their choice. In addition to the URLs dispersed throughout the text, the Expanded Applications can be accessed by going to the Expanded Applications home page through this URL: bit.ly/2yoUCOY. Note that when you enter URLs for the Extended Applications, take care to distinguish the following characters:

The following features highlight the flavor of computing technology that distinguishes much of our exposition.

• Almost 600 computer-generated figures show students vivid pictures of direction fields, solution curves, and phase plane portraits that bring symbolic solutions of differential equations to life.

• About three dozen application modules follow key sections throughout the text. Most of these applications outline technology investigations that can be carried out using a variety of popular technical computing systems and which seek to actively engage students in the application of new technology. These modules are accompanied by the new Expanded Applications website previously detailed, which provides explicit code for nearly all of the applications in a number of popular technology platforms.

• The early introduction of numerical solution techniques in Chapter 2 (on mathematical models and numerical methods) allows for a fresh numerical emphasis throughout the text. Here and in Chapter 7, where numerical techniques for systems are treated, a concrete and tangible flavor is achieved by the inclusion of numerical algorithms presented in parallel fashion for systems ranging from graphing calculators to Matlab and Python.