Supplementing the rule of three with the physical perspective leads to most of the principal ingredients of mathematical modeling: deriving the model, analyzing the resulting mathematical problem, and interpreting the results in light of the original physical problem [8]. While some might resist including all three steps in an introductory differential equations course, the last step, interpreting analytical results in a physical context, is certainly central to any such course.
These kinds of problems can ask directly about the behavior predicted by a specific model, or they can ask about generic behavior:
Beginning in 1847, many residents of Ireland left the island to escape the terrible potato famine. The population of Ireland at that time could be modeled by P' = 0.015P - 0.209, P(1847) = 8. (P is population in millions; time is measured in years.) Will the population grow or decline?Suppose y' = y + b, b >0, is a model of some physical process. Does the part of the process modeled by the term b act to raise or lower y? What if b < 0?
Questions like these are harder than they look. To ease the door open in the context of the population question, for example, one could also ask
Is the equation P' = 0.015 P - 0.209 a reasonable model of growth coupled with emigration? Begin your answer by determining the units of each term. What does each term represent? What are the units of the coefficients 0.015 and 0.209 and what do they represent? Are these values reasonable? How do they compare with those for the U.S.? For China? Are the values given here consistent with claims like, ``Birth rates of human populations seldom exceed 4%.''? What is the connection between birth rate and the number of surviving children born to each woman?
In many ways, the modeling approach to teaching differential equations is the culmination of developing a graduated spectrum of challenging exercises. Besides offering students obvious links to their studies in other disciplines, modeling encompasses all of the principles demonstrated here for developing better exercises.
Physical problems are naturally ambiguous, for it is seldom obvious which questions are reasonable and which are not. Furthermore, the mathematics is often too well hidden at the start to offer useful clues.
Using mathematics to understand a physical process always demands asking about; such problems never ask for the use of a particular method.
Physical problems demand mastery of the mathematical vocabulary in order to marshal the proper tools for analysis. These problems also demand mastery of a minimal vocabulary in the target discipline, demonstrating that technical vocabulary is not a demand peculiar to mathematics.
Finally, a good understanding of a mathematical model usually demands full use of all of the perspectives of the rule of three, algebraic, graphical, and numerical.