Teaching Differential Equations with Modeling and Visualization

Abstract

In this article, I explain the history of using Interdisciplinary Lively Applications Projects (ILAPs) in an ordinary differential equations course. Students want to learn methods to “solve real world problems,” and incorporating ILAPs into the syllabus has been an effective way to apply solution methods to situations that students may encounter in other disciplines. Feedback has been positive and will be shared. Examples of ILAPs currently used will be referenced. For more information about how to develop ILAPs, see Huber and Myers (in Innovative Approaches to Undergraduate Mathematics Courses Beyond Calculus, 2005). 1 Confident and Competent Problem Solvers One of my goals each semester is to develop students into confident and competent problem solvers. The competence comes from practice, solving problem after problem. The confidence also comes with experience, from knowing what to do when facing a new problem. Many textbooks give students an equation and ask them to solve it. In the past, if the textbook gave students the harmonic oscillator equation, complete with values for the mass, damping coefficient and spring constant, students could put in the values and then use the appropriate technique to solve for the displacement or velocity of the mass. However, when confronted with a word problem and asked to develop the model’s differential equation, students struggled. Students were comfortable with their “Plug and Chug” method of solving, as long as they knew what and where to plug. That got me thinking: What if we just gave them the forces acting on the problem, to include any non-homogeneous driving function, and asked them to predict long-term behavior? A few years ago, as an experiment suggested by my colleague Don Small at West Point, I gave a group of students a differential equations problem in the form of a long word problem and asked them to set up the model (define variables, state what is given, make a few valid assumptions, write down what they were asked to determine, and explain the technique they would use to solve it). Suppose that the growth of an alligator population in the Okefenokee Swamp is proportional to its population at a rate of 0.05 per year, and that the initial population is 7200 gators. Alligator populations have risen to such CODEE Journal http://www.codee.org/ high levels that the U.S. Fish and Wildlife Service “harvests” gators by allowing tightly controlled hunting of 650 to be hunted in one year, followed by no harvesting in each of the next two years of these animals, famous for their durable hide and excellent meat. This harvesting strategy is repeated every three years. Model the IVP and predict how many gators will there be after 6, 7, and 8 years. Further, what would the harvesting need to be to wipe out the gator population? I told them NOT to solve it, unless they had time. We had already covered using Laplace transforms to solve these types of problems, but I wanted to see if they could establish the problem to be solved. Afterwards, I asked the students to explain their problem-solving process, to include any assumptions they had to make. The real value of this exercise came from the explanations. Some students wrote about their anxieties about having a “vague” problem, while others wrote about their happiness in being able to set up a “real-world” problem. The harvesting scheme involves a Heaviside function which is shown in Figure 1 below. The horizontal axis shows time in years and the vertical axis shows the number of gators hunted. The solution to the problem should show a series of population increases and decreases, due to the cyclic harvesting. A plot of the particular solution is shown in Figure 2. 1 2 3 4 5 t 6 7 8 9 200 400 600 Figure 1: Harvesting Gators in the Okefenokee Swamp 1 2 3 4 5 t 6 7 8 9 7000 7500 8000 8500 Figure 2: Gator Population in the Okefenokee Swamp