Using a Model to Give a Grand Tour of a First Course in Differential Equations
0Presented by:
Samuel Graff, John Jay College of Criminal Justice, CUNY, New York NY USA
https://qubeshub.org/community/groups/simiode/expo
Abstract: Formulating a versatile population growth model affords an opportunity to survey some of the important concepts that are presented during a first course in ordinary differential equations. Initially, the classical Malthusian law allows for a discussion of first order linear differential equations including the notion of an integrating factor. While the full model is nonlinear with respect to the dependent variable, it can be solved explicitly, yielding an implicit representation of the solution using the separation of variables method.
The implicit representation of the solution suggests that an analysis of the long-term behavior of all solutions might be challenging. Fortunately, the fact that there are three equilibrium solutions offers a gateway to the geometrical theory of differential equations. In this context, the phase line, the stability of the equilibria, and the long-term behavior of the entire family of solutions may be explored.
Students are often spellbound by the geometric analysis and the reality that so much information can be obtained just from the differential equation. For many students, this experience offers a compelling answer to their query concerning the utility of calculus itself.