% SIMIODE-TeacherStudent-Template.tex and simiode.cls for LaTeX2e
% Version 22 November 2016.
%
% This template is to be used with the class file "simiode.cls".
%
% The latest version of this class and template files
% can be found at
%
% in the Author section.
%
% A "Quick Start Guide" appears below.
%
% This file must be processed with version LaTeX2e or higher.
% Appreciation to Mark Frantz, IUPUI; John Thoo, Yuba College; and Eric Sullivan, Carroll College.
%
\documentclass{simiode}
%
% With this version of the tex file and the associated SIMIODE.cls file you can produce BOTH a Student Version and a Teacher Version of your modeling scenario or technique narrative.
%
% Student Version contains the title STUDENT VERSION along with the NAME OF SCENARIO, with no abstract, no keywords, no tags, no author identifiers, and no COMMENTS section. The Student Version contains only the STATEMENT of the modeling scenario only with appropriate figures, data, instructions, and perhaps a bibliography appropriate for student resourcing, but nothing in the bibliography which would "give away" the nature of the modeling scenario, etc.
%
%The Teacher Version contains the material in the Student Version, namely the title NAME OF SCENARIO, now headed with TEACHER VERSION, but also full author identification, abstract, key words, tag words, and, following the STATEMENT, a section of COMMENTS which can be discussion of presentation ideas, "solutions," techniques, etc.
%
% Authors can prepare a full Teacher Version using the notions immediately below to create a pdf for review, but in review please eliminate all information about the author as we use a double-blind referee system. Such information will be restored when accepted and published in SIMIODE at www.simiode.org.
%
%*********** Toggle the following line to turn the teacher version on (1) or off (0). ***********
%
\def\TeacherEdition{0} % Toggle this line's value.
\newcommand{\teacher}[1]{\ifnum\TeacherEdition=1 #1 \fi}
%
% The \teacher{ } command wraps any text that will appear only in the teacher version of
% the document. See this file for examples. This has been done for you in the document below.
% Rename the pdf after you compile each final version.
%
\usepackage{amsmath,amssymb}
\usepackage{color}
\usepackage{graphicx}
\usepackage{hyperref}
\usepackage{url}
%
\renewcommand{\baselinestretch}{1.3}
%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% NOTE regarding macros %%
%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Any macros defined for your paper should be contained in
% the top matter. Likewise, any environment definitions such
% as \newtheorem or \newenvironment should also go in the
% top matter.
%
% Do not \input your macros as separate files within your
% final version because more files make it harder for the
% editors and users to keep track of and/or modify and
% append your paper.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Top matter is above and
% the beginning of the document below.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}
\newpage
% You only need to change the NAME OF SCENARIO in the next line and delete the footnote. See above for instructions on toggling from Teacher Version to Student Version.
\title{\ifnum\TeacherEdition=1 TEACHER VERSION \else STUDENT VERSION \fi\\CONJECTURING SOLUTIONS FOR LINEAR SYSTEM OF DIFFERENTIAL EQUATIONS}
\markboth{Conjecturing Solutions Linear Systems}{Conjecturing Solutions Linear Systems}
\author{Brian Winkel\\ Director
SIMIODE\\
Chardon OH 44024 USA\\ BrianWinkel@SIMIODE.org}
\makeStitlePDFLaTex
\begin{abstract}
We lead students from the solution for $y(t) = k\, y(t)$ to a natural extension for the solution conjecture for a system of two constant coefficient, homogeneous, linear differential equations introducing eigenvalues and eigenvectors through student discovery.
\end{abstract}
\section*{INTRODUCTION}
We consider a system of linear differential equations
\begin{eqnarray}\label{eq:system}
x'(t) &=& a \, x(t) + b \,y(t) \\ \nonumber
y'(t) &=& c \,x(t) + d \,y(t)
\end{eqnarray}
with $x(0) = x_0$ and $y(0) = y_0$.
We can put this in matrix form:
\begin{equation}\label{eq:matrix}
\left[
\begin{array}{c}
x'(t) \\
y'(t)
\end{array}
\right] = \left[
\begin{array}{c}
a \, x(t) + b \,y(t) \\
c \,x(t) + d \,y(t)
\end{array}
\right] = \left[
\begin{array}{cc}
a&b \\
c&d
\end{array}
\right] \cdot \left[
\begin{array}{c}
x(t) \\
y(t)
\end{array}
\right]
\end{equation}
with $x(0) = x_0$ and $y(0) = y_0$.
OR
$$\displaystyle{X'(t) = A\cdot X(t)} \, , $$
where
\begin{equation}\label{eq:Xarray}
X'(t) = \left[ \begin{array}{c}
x'(t) \\
y'(t)
\end{array}\right] \quad , \quad A = \left[
\begin{array}{cc}
a&b \\
c&d
\end{array}
\right] \quad ,\quad X(t) = \left[ \begin{array}{c}
x(t) \\
y(t)
\end{array}\right]
\end{equation}
$X'(t) = A\cdot X(t)$ is very suggestive for a solution of the form $X(t) = c e^{A t}$. Thus one could conjecture a solution and try it out to see what $c$ would have to be, but we have to remember $A$ is a matrix, not a number, and while there are theories of what $e^A$ would mean for a matrix $A$ we choose not to go there now.
Instead, we conjecture a vector solution for $X(t)$ which has the best of both worlds (1) nice exponential form with no matrices in the exponent and (2) vector status where we can see the pieces of the solution:
\begin{equation}\label{eq:vecsol}
X(t) = \left[ \begin{array}{c}
x(t) \\
y(t)
\end{array}\right] = \left[ \begin{array}{c}
u \\
v
\end{array}\right] \cdot e^{\lambda t}\, .
\end{equation}
So here we are suggesting $x(t) = u e^{\lambda t}$ and $y(t) = v e^{\lambda t}$.
\subsection*{Assignment}
\begin{enumerate}
\item Now put the conjecture (\ref{eq:vecsol}) in (\ref{eq:Xarray}) and see what has to happen to $a$, $b$, $c$, $d$ and $u$, $v$ in order to obtain solutions - interesting solutions, i.e. other than $u = 0$ and $v=0$. Go for it!!! Summarize your conclusions in a nice write-up
\item Illustrate the solution strategy from assignment (1) using the following system
\begin{eqnarray}
x'(t) &=&-4 \, x(t) -2\,y(t) \\ \nonumber
y'(t) &=& -1 \,x(t) -3 \,y(t)
\end{eqnarray}
with $x(0) = 2$ and $y(0) = 1$.
\end{enumerate}
\teacher{
\section*{NOTES FOR TEACHERS}
We have used this approach successfully for years. When asked, students substitute conjecture (\ref{eq:vecsol}) into the system of differential equations (\ref{eq:system}) and produce necessary conditions on $a$, $b$, $c$, $d$ and $u$, $v$ for a solution to exist from the resulting equations with the $e^{\lambda t}$ term canceled out from every term:
\begin{equation}\label{eq:cancel}
\left[
\begin{array}{cc}
a -\lambda & b \\
c & d -\lambda
\end{array}
\right] \cdot \left[
\begin{array}{c}
x(t) \\
y(t)
\end{array}
\right] = \left[
\begin{array}{c}
0 \\
0
\end{array}
\right]
\end{equation}
Students then see the only non-trivial or non-zero solutions which can occur is when in non-matrix form we have the following:
\begin{eqnarray}\label{eq:nonmat}
(a - \lambda)\,x(t) + b \,y(t) &=& 0\\ \nonumber
c \,x(t) + (d - \lambda) \,y(t) &=& 0\, .
\end{eqnarray}
This means plot of both equations (being lines) in the $x(t)$-$y(t)$ plane have to be the same, i.e. they go through the origin and have identical slopes,
\begin{equation}\label{eq:slopes}
\frac{-(a - \lambda)}{b} = \frac{-c}{(d - \lambda)}\, .
\end{equation}
This implies that solutions to the original differential equation, (\ref{eq:system}), must satisfy the following quadratic equation:
\begin{equation}\label{eq:char}
\lambda^2 - (a+d)\lambda+ a d - b c = 0 \, .
\end{equation}
This equation characterizes the solutions of (\ref{eq:system}) and so is called the {\it characteristic equation\/} for the system (\ref{eq:system}) and the roots of this equation, $\lambda_1$ and $\lambda_2$, are called {\it eigenvalues} and their corresponding vectors $\left[\begin{array}{c}u_i\\ v_i\end{array}\right]$, $i = 1, 2$, are called {\it eigenvectors}.
\subsection*{Practical Issues}
When teaching this material using by-hand analysis we do not dwell on all the cases of the roots of a quadratic (i.e. real and unequal, real and equal, and complex), but rather we confine this discussion to real and unequal roots, always with negative eigenvalues as no real system modeling in this way can have positive eigenvalue. We study the cases, real and equal and complex eigenvalues, shortly thereafter in context of a modeling experience and use technology to obtain the corresponding eigenvalues and eigenvectors as well as handle the complexities of the computation. We engage in appropriate algebra and graphical feedback to relate the behavior of the solution to the nature of the eigenvalues.
With the littlest of prompting students can actually derive eigenvalues and eigenvectors without ever having seen a definition or application. Indeed, this approach is one of discovery in which students actually find these objects, but may not have a name for them. However, they sure do have a reason for knowing about them as they see them (they actually found them!) to be the building blocks for at least a 2 by 2 system of linear differential equations.
This approach NEVER FAILS to engage students and draw them in to discovery of something very useful - eigenvalues and eigenvectors.
}
\end{document}