Visualization of solutions to certain elementary differential equations
on the TI-85

John F. Lucas

Serious treatment of first and second-order linear differential equations is a relatively new focus that appears in the second calculus course in various reform curricula. This paper specifically addresses the graphical perspective of the Calculus Consortium at Harvard (CCH), using technology of the Texas Instruments TI-85 graphics programmable calculator. Considerable insight about the nature and solution of differential equations can be afforded by students using a combination of programs and the TI-85 DifEq graphing option. We investigate five different solutions. First, we examine an algebraically-presented equation (drug-injection model) from which students produce a differential equation, then a slope-field general solution, and finally a DifEq-presented specific solution which can be superimposed in the slope field and checked by drawing in the known algebraic solution. After that, we help solve a murder mystery using Newton's Law of Cooling and a "negative-incremented" time dimension, using the trace feature to approximate the time of the murder. The next two applications treat systems of first-order differential equations --an S-I-R model of an epidemic and a predator-prey model with slope field, trajectory and time-series analyses. The last example involves damped oscillation in a spring-mass system, where the solution curve is drawn first and then estimates are used to approximate its dampening and oscillatory functions.

At the University of Wisconsin-Oshkosh, we have been teaching all sections of our mainstream calculus courses using both the Harvard Calculus Consortium (CCH) curriculum and graphics programmable calculators (TI-85) since fall of 1992. We selected the CCH materials because we were searching for a breath of fresh air by way of calculus reform, and the Harvard calculus emphasized a fusion of various perspectives --graphical, numerical, symbolic, and verbal-- in the context of non-standard, interesting problems. The TI-85 was a beautiful technological match for the curriculum because of its accessibility, portability, cost, and variety of menu options involving graphical and numerical options. We have not been disappointed by either choice. At least a dozen of our faculty have taught one course in our three-semester sequence, and five of us have taught the entire three-semester sequence which includes multivariable calculus.

This paper is focused on elementary differential equations, which comprise about 40% of the second-semester CCH course. The TI-85 has a menu (DifEq) that graphs solutions quite nicely for the kind of first- and second-order differential equations appearing in the Harvard Calculus, allowing students excellent visual accessibility for analysis, interpretation and, in some (but not all) cases, comparison with algebraic solutions. Several ancillary programs on Euler's method (Numerical, Graphical and Systems of Differential Equations), Trajectories, Slope Fields, and Time Series were written to assist the student with numerical and graphical work on differential equations.

We will examine here five problems (four of which are taken directly from the CCH text) that illustrate the flavor of the curriculum and the application of the TI-85 to assist learning, particularly visual learning. These situations include an abstract differential equation (drug-injection model), an S-I-R epidemic model, a murder mystery (using Newton's Law of Cooling), a predator-prey system (robins and worms), and a linear harmonic oscillator (damped spring-mass system).

John F. Lucas
Lucas - 2 OCT 1996



There is a PDF file available for this paper.


Gómez, P. & Waits, B. (Eds.) (1996). Roles of calculators in the classroom.

Mail comments to Pedro Gómez: pgomez@uniandes.edu.co