Overview and structure

This text uses several complementary, spiral approaches.

• Models of physical problems motivate differential equations. The differential equations demand analytical, numerical, and graphical tools for their analysis. Interpreting the results of the analysis leads back to the physical problem, which then demands deeper analysis or better models that themselves require more sophisticated analysis.

  These distinct steps are often identified explicitly by section titles or by marginal labels like Model, Analysis, or Interpretation. For example, subsections 2.1.3, 2.1.4, and 2.1.5, p. 30-34, successively derive the simple population model, analyze it, and interpret the analysis; the subsection titles are A Model, Analysis, and Interpretation, respectively. Marginal notes on p. 2-3 identify the modeling, analysis, and interpretation steps for the projectile model considered there.

• Most mathematical ideas are introduced intuitively before they are defined formally, so that definitions can arise naturally, rather than appear as arbitrary rules.
• To give students the experience of generalization, common methods and ideas are introduced in succession for first-order scalar equations, then for second-order equations, then for first-order systems. For example, characteristic equations appear in all three settings, and each reappearance of the problem of solving a constant-coefficient, homogeneous, linear equation turns back to a previous, simpler problem for guidance in attacking the newer, more complex problem.
  • []course, the mathematician in me is impatient to show the power of the general approach. However, I have found that such haste keeps all but the best students from seeing the power of generalization. Climbing the mountain step-by-step gives a better appreciation for the view than traveling to the summit for the first time via helicopter!

Analytical, graphical, and numerical tools are all introduced early in anticipation of later refinement, extension, and generalization. Chapter 1, Prologue, surveys this mix, giving a sample of modeling, of finding an analytic solution of an initial-value problem via separation of variables, of a numerical approximation using Euler's method, and of direction fields and solution graphs. Subsequent chapters are more specialized, as summarized in table 6.

• []approach could be viewed as the ``Rule of Three Plus One'', the analytical, graphical, and numerical perspectives augmented by the physical. See P. W. Davis, Asking Good Questions about Differential Equations, College Math. J., 25(5) 1994, 394-400.