Lab 7

This lab is a step away from the sciences, in fact it uses fiction to explore how ODE systems can be used to express relationships. It also uses fiction to show how small changes in the system can be used to express changes in the problem. To do this we will look at the problem as a three act play.

I am going to give you a system of ODE's with instructions as to how to modify it to express 2 different changes in the problem and a comment or two as to what caused the changes. You are to graph the original and each of the modified systems in 3 ways: MATH The window for $y=y(x)$ must be at least $\ -5\leq x\leq 5,$ $-5\leq y\leq 5$ and the windows for the other 2 $0\leq t\leq 60,$ MATH You are also to create a paragraph or 2 for each of the 3 systems that would serve as a summary of each act. The play:

ROMEO and JULIET

in 3 acts with apologies to Shakespear

Prologue

The lover's opening statements:

Romeo: ''When I first saw Juliet I was attracted to her (x(0)=2) but my love for her decreases in proportion to her love for me!''

Juliet: ''At first Romeo was just one of the boys (y(0)=0) however my love for him grows in proportion to his love for me!''

To reflect these statements mathematically consider the following:

t is time, $0\leq t\leq 60$ days

$\qquad x(t)$ is Romeo's love for Juliet

$\qquad y(t)$ is Juliet's love for Romeo

Scale the $x$ and $y$ axis as follows:

Hysterical Sweet Ecstatic

Hatred Disgust Indifference Attraction Love

-5 -2.5 0 2.5 5

ACT 1

The ODE system for the opening statements is MATH

Run these to find out why Juliet went to a psychologist.

ACT 2

Using a tranquilizer the psychologist causes a change in Juliets emotions and therefore her ODE equation.

She reins in her feelings. To show this add the term $-0.1y$ to Juliet's equation.

Run this new system with $x(0)=2,$ $y(0)=0$ and figure out why they both went together to a new psychologist. Was this necessary?

ACT 3

The new psychologist tells Juliet to drop the tranquilizer (back to the original, act 1, equations) and teaches them to have more control over their emotions changing their equations in a different manner. The changes: replace $x$ with $x-2$ and $y$ with $y-2\ .$

This is the last act so run these equations, again $x(0)=2,$ $y(0)=0,$ and try to predict the future for Romeo and Juliet. Is it good or bad?

adapted from an article in The College Mathematics Journal by J M McDill and Bjorn Felsager.

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