My preferences

My own preferences for models to teach are:

• scalar population models (sections 2.1-2.2) because they are simple to derive, they are easy to understand without much background in the physical sciences, and they motivate all of the important first-order analytical, numerical, and graphical concepts,
• the heat-flow model (section 2.3) because it is easy to argue that it is plausible (although the underlying physics is subtle), it has an intuitively obvious stable steady state, and it seems more ``real'' to most students of science and engineering,
• spring-mass and pendulum models (sections 5.1-5.2) because these devices are easy to demonstrate in class, they introduce oscillatory phenomena, and they provide a complete foundation for most of the elementary two-dimensional ideas, particularly linearization and the phase plane,
• SIRS (or any other multiple species model from section 2.4) because for beginners the underlying science is simple and the phase plane is interesting,
• diffusion models (sections 9.1 and 9.4) because are an important class of boundary-value and initial-boundary-value problems for ordinary and partial differential equations, leading to important analytical and numerical ideas.

In class, I derive the population models in sections 2.1 and 2.2 (Malthus, emigration, logistic), argue for the plausibility of the heat-flow model (section 2.3), demonstrate a vertical spring-mass system and derive its governing equation (section 5.1), and derive one of the diffusion models in section 9.1. I use ``Take my word for it'', augmented by a little hand waving and a reading or a homework assignment, to introduce as needed the multiple species models (predator-prey, SIRS, competition) of section 2.4. The linear and nonlinear pendulum equations (section 5.2) enter with slightly more ceremony and homework emphasis but usually without a formal derivation.

My preferences among the mathematical ideas and methods are

• graphical concepts such as direction fields, phase planes, and sketching solution graphs from differential equations because they reinforce the rate of change concepts fundamental to calculus,
• stability and linearization because stability is an intuitive concept with natural analytic and graphical interpretations, its analysis often requires linearization (perhaps the most ubiquitous process, imperfections notwithstanding, in science and engineering), and linearization leads immediately back to the derivative and to Taylor's theorem.
• elementary numerical methods because they have natural graphical interpretations, they involve linearization, they call upon Taylor's theorem, and their sophisticated descendants are so important in practice,
• elementary analytical methods because they exercise manipulative skills, they provide a ``plug 'n' chug'' refuge for the student struggling with more difficult open-ended problems, and their limitations illustrate the importance of understanding. (To paraphrase Peter Hammer, ``The purpose of differential equations is insight, not formulas (or numbers or graphs).'')