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Differential Equations in the Calculus Courses

Many of the applications of calculus come through differential equations which model real life phenomena. In the very first calculus course, students in our courses wrestle with qualitative descriptions of models governed by differential equations. You'll see in this first example, that students simply need a good grasp of the fact that the differential equation is simply a way of measuring the growth rate of a function. Using the fact is not a skill to be taken lightly. Here's an early example from the course on derivatives.

You get a reasonable interpretation of the logistic differential equation

y'[t] = a y[t] (1 - y[t]/b) with y[0] between 0 and b

by imagining that y[t] is the number of catfish in a given lake on a catfish farm a weeks after the lake was stocked with a catfish. As time goes on, the catfish population increases until it reaches its steady-state population of b catfish. But the catfish farmer doesn't grow catfish as pets; the farmer is in business to harvest catfish and to sell them so that hungry persons can fry them up and then wash them down with a couple of cold beers or iced teas. Measure time in weeks and assume the farmer wants to harvest r fish per week and explain why

y'[t] = a y[t] (1 - y[t]/b) - r

lays the base for a reasonable model.

In the second part of the course, that part on integrals, the problem returns and students are expected to find solution formulas. It pops up again in the Differential Equations course itself. There we expect much deeper analysis, including attempts at spotting bifurcations.