Backwards and forwards through a differential equation

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

Backwards and forwards through a differential equation

This example is introduced in class after students have explored easy (guessable) differential equations, such as

a. with (or )

b. with (or some other boundary condition).

At this point, students have familiarity with slope fields and the DifEq menu of the TI-85 (where the dependent variable is Q1 and the independent variable is t).

They are given the equation

: (if , )

and asked to differentiate, obtaining

Now, they replace

with

from the original equation, obtaining a differential equation of the form

(we already know the specific solution for this differential equation if the boundary condition is

. It's

Next, we explore the situation visually in two ways:

a. Use the Slope field program on the TI-85 (see Appendix) to construct a slope field over the window .
Save the slope field picture by selecting STPIC and naming it DE1.

b. Next, set up the DifEq graphing menu, as follows:

Q'1= (Q1/t)*(1-0.2t)
RANGE setting , t-step .05, tplot 1 (plotting starts at t value of 1), and
INITC QI1=8.19 (this is the value of Q1,or "y", when t (or "x") is 1)
GRAPH the solution curve, we see the window shown in Figure 1.
Figure 1.
Algebraically, check the solution against our graph by selecting DRAW, then DRAWF and typing in 10xe^(-0.2x). The drawn curve will start at (filling in the gap where ) and thereafter follow very closely the path of the solution curve.

c. Finally, superimpose the slope field on top of the DifEq solution curve by selecting RCPIC from the menu and pressing the F- key corresponding to DE1. We are in effect, viewing a trajectory in a slope field, shown in Figure 1.

Figure 2.

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