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chap2-firstorder.tex
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chap2-firstorder.tex
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% Chapter 2 - First-Order ODEs
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{topic}[First-Order Differential Equations]
\vfil
\begin{center}
\begin{minipage}{200pt}
\includegraphics*[width=200pt]{images/chap2-xkcd.png}
\hfill {\footnotesize (image from \href{https://www.xkcd.com/793/}{xkcd - comic \#793})}
\end{minipage}
\end{center}
\end{topic}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% MODULE - Introduction
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{module}{Introduction to Differential Equations}
% \Title{Definition}
\label{ODE:intro}
% \Heading{Textbook}
% \Heading{Objectives}
% \begin{itemize}
% \item Bla bla bla
% \end{itemize}
%
% \Heading{Motivation}
\begin{lesson}
\Title{Introduction to Differential Equations}
% \Heading{Objectives}
% \begin{itemize}
% \item The second step in Mathematical modelling is to construct a representation of how the team will be attempting to solve the problem.
% \item Create a mind map of the problem. This is a structured way to brainstorm possible solutions and their requirements.
% \end{itemize}
%
% \Heading{Motivation}
%
%\begin{annotation}
% \begin{goals}
% \Goal{Extra Reading}
% Math Modelling: Getting started and getting solutions, Bliss-Fowler-Galluzzo
%
% \hfill \qrcode{https://m3challenge.siam.org/resources/modeling-handbook}
% \end{goals}
%\end{annotation}
% \Heading{Extra Reading} \href{https://m3challenge.siam.org/resources/modeling-handbook}{Math Modelling: Getting started and getting solutions, Bliss-Fowler-Galluzzo}
%
\end{lesson}
\input{modules/module07-intro.tex}
\input{modules/module07-intro-noexercises.tex}
\end{module}
\newpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% MODULE - Solutions
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{module}{Solutions of Differential Equations}
%\Title{Solutions}
\label{ODE:solutions}
\input{modules/module08-sols.tex}
\input{modules/module08-sols-exercises.tex}
\end{module}
\begin{lesson}
\Title{Solutions of Differential Equations}
\Heading{Textbook}
\begin{itemize}
\item Modules 7, 8
\end{itemize}
\Heading{Objectives}
\begin{itemize}
\item Identifying the order of a differential equation
\item Identifying a linear vs nonlinear differential equation
\item Knowing how to check if a function is a solution of a differential equation
\end{itemize}
\Heading{Motivation}
This is an introduction to differential equations.
Students have different levels of experience with differential equations. We want to establish a common notation.
\begin{annotation}
\begin{goals}
\Goal{Using pre-class ODEs.}
\begin{itemize}
\item At the start of class, put about 8 student ODEs on the board
\item Get students to identify them: order + linearity
\item Get some info about the ODEs to the whole class
\end{itemize}
\end{goals}
\end{annotation}
\Heading{Preparation for Class}
\begin{itemize}
\item Students should bring an ODE of their choice to class with some information about it
\item Read textbook modules.
\end{itemize}
\Heading{Tutorials and Projects}
\begin{itemize}
\item Project \ref{proj:fishery}: \fisherytitle
\end{itemize}
\end{lesson}
%\newpage
\begin{annotation}
\begin{goals}
\Goal{Without solving.}
\begin{itemize}
\item Students should try to answer this question without solving the differential equation.
\item Check properties of the ODE:
\begin{itemize}
\item What does $x^2-1=0$ mean for the solution?
\item When is $y'>0$? What does that mean for the solution?
\item When is $y'<0$? What does that mean for the solution?
\end{itemize}
\end{itemize}
\end{goals}
\end{annotation}
\question
Which of these shows solutions of $y' = (x-1)(x+1) = x^2 - 1$ ?
\newlength{\len}
%\setlength{\len}{120pt}
%\begin{tabular}{ccc}
%\includegraphics[width=\len]{images/module8-figs-6-small.png}
% & \includegraphics[width=\len]{images/module8-figs-3-small.png}
% & \includegraphics[width=\len, page=2]{images/module8-figs-2-small.png} \\
%A & B & C \\[15pt]
%%
%\includegraphics[width=\len]{images/module8-figs-1-small.png}
% & \includegraphics[width=\len]{images/module8-figs-5-small.png}
% & \includegraphics[width=\len]{images/module8-figs-4-small.png} \\
%D & E & F \\
%\end{tabular}
\def\modeightA{
\begin{tikzpicture}[scale=0.65,yscale=0.8]
\begin{scope}
\clip (-3,-3) rectangle (3,3);
\foreach \k in {-9,-7, ..., 31} {
\draw[samples=50,domain=-3:3,variable=\x,color=gray!80!black] plot ({\x},{(\k-3*(\x*\x))/2});
}
\end{scope}
\draw[thick] (-3,-3) -- (-3,3);
\draw[thick] (-3,-3) -- (3,-3);
\foreach \k in {-3,-2, ..., 3} {
\draw ({\k,-3}) node[below] {\tiny $\k$};
\draw ({-3,\k}) node[left] {\tiny $\k$};
}
\end{tikzpicture}
}
\def\modeightB{
\begin{tikzpicture}[scale=0.65,yscale=0.8]
\begin{scope}
\clip (-3,-3) rectangle (3,3);
\foreach \k in {-9,-7, ..., 39} {
\draw[samples=50,domain=-3:3,variable=\x,color=gray!80!black] plot ({\x},{((\x*\x*\x)/3-(\x*\x)+\k/2)});
}
\end{scope}
\draw[thick] (-3,-3) -- (-3,3);
\draw[thick] (-3,-3) -- (3,-3);
\foreach \k in {-3,-2, ..., 3} {
\draw ({\k,-3}) node[below] {\tiny $\k$};
\draw ({-3,\k}) node[left] {\tiny $\k$};
}
\end{tikzpicture}
}
\def\modeightC{
\begin{tikzpicture}[scale=0.65,yscale=0.8]
\begin{scope}
\clip (-3,-3) rectangle (3,3);
\foreach \k in {-15,-13, ..., 29} {
\draw[samples=50,domain=-3:3,variable=\x,color=gray!80!black] plot ({\x},{-(((\x+1)*(\x+1)*(\x+1))/3-((\x+1)*(\x+1))+\k/2)});
}
\end{scope}
\draw[thick] (-3,-3) -- (-3,3);
\draw[thick] (-3,-3) -- (3,-3);
\foreach \k in {-3,-2, ..., 3} {
\draw ({\k,-3}) node[below] {\tiny $\k$};
\draw ({-3,\k}) node[left] {\tiny $\k$};
}
\end{tikzpicture}
}
\def\modeightD{
\begin{tikzpicture}[scale=0.65,yscale=0.8]
\begin{scope}
\clip (-3,-3) rectangle (3,3);
\foreach \k in {-15,-13, ..., 29} {
\draw[samples=50,domain=-3:3,variable=\x,color=gray!80!black] plot ({\x},{(((\x+1)*(\x+1)*(\x+1))/3-((\x+1)*(\x+1))+\k/2)});
}
\end{scope}
\draw[thick] (-3,-3) -- (-3,3);
\draw[thick] (-3,-3) -- (3,-3);
\foreach \k in {-3,-2, ..., 3} {
\draw ({\k,-3}) node[below] {\tiny $\k$};
\draw ({-3,\k}) node[left] {\tiny $\k$};
}
\end{tikzpicture}
}
\def\modeightE{
\begin{tikzpicture}[scale=0.65,yscale=0.8]
\begin{scope}
\clip (-3,-3) rectangle (3,3);
\foreach \k in {-9,-7, ..., 31} {
\draw[samples=50,domain=-3:3,variable=\x,color=gray!80!black] plot ({\x},{-(\k-3*(\x*\x))/2});
}
\end{scope}
\draw[thick] (-3,-3) -- (-3,3);
\draw[thick] (-3,-3) -- (3,-3);
\foreach \k in {-3,-2, ..., 3} {
\draw ({\k,-3}) node[below] {\tiny $\k$};
\draw ({-3,\k}) node[left] {\tiny $\k$};
}
\end{tikzpicture}
}
\def\modeightF{
\begin{tikzpicture}[scale=0.65,yscale=0.8]
\begin{scope}
\clip (-3,-3) rectangle (3,3);
\foreach \k in {-9,-7, ..., 39} {
\draw[samples=50,domain=-3:3,variable=\x,color=gray!80!black] plot ({\x},{(-(\x*\x*\x)/3+(\x*\x)-\k/2)});
}
\end{scope}
\draw[thick] (-3,-3) -- (-3,3);
\draw[thick] (-3,-3) -- (3,-3);
\foreach \k in {-3,-2, ..., 3} {
\draw ({\k,-3}) node[below] {\tiny $\k$};
\draw ({-3,\k}) node[left] {\tiny $\k$};
}
\end{tikzpicture}
}
\begin{instructoronly}
\hspace{-1cm}
\end{instructoronly}
\begin{tabular}{ccc}
\modeightA
& \modeightB
& \modeightC \\
A & B & C \\[15pt]
%
\modeightD
& \modeightE
& \modeightF \\
D & E & F \\
\end{tabular}
\begin{annotation}
\begin{goals}
Get students to figure out this core exercise in two different ways:
\begin{itemize}
\item by solving the ODE
\item without solving the ODE
\end{itemize}
\end{goals}
\end{annotation}
\bookonlynewpage
\begin{annotation}
\begin{goals}
Quick exercise
\end{goals}
\end{annotation}
\question
We seek a first-order ordinary differential equation
\quad $y' = f(x)$ \quad
whose solutions satisfy
$$
\begin{cases}
y(x) \mbox{ is increasing if } x<2 \\
y(x) \mbox{ is decreasing if } 2 < x < 4 \\
y(x) \mbox{ is increasing if } x > 4
\end{cases}
$$
%
Write down or graph a function $f(x)$ that would produce such solutions.
\bookonlynewpage
\begin{annotation}
\begin{goals}
\Goal{Stronger students.}
The 5$^{\rm th}$ part is for stronger students to think about while they wait for the others to finish.
Will be addressed later in module 13 (Properties of solutions). %\ref{ODE:properties}
They can see that:
\begin{itemize}
\item If $y(t_0)>0$, then $y(t)>0$ for $t>t_0$
\item If $y(t_0)<0$, then
\begin{tikzpicture}
\draw[thick,-{\seta}] (-0.25,0) -- (3.5,0) node[above] {\small $t$};
\draw[thick,-{\seta}] (0,-2) -- (0,2) node[left] {\small $y$};
\draw[samples=20, ultra thick,domain=-1.25:1.25,smooth,variable=\x,cyan] plot ({\x+1.5},{\x*\x*\x});
\draw (3,1) node[right,cyan] {\Huge ?};
\end{tikzpicture}
\end{itemize}
\end{goals}
\end{annotation}
\question
Consider the ODE \quad $y'(t) = \big(y(t)\big)^2$ \quad .
Which of the following is true?
\begin{parts}
\item $y(t)$ must always be decreasing
\item $y(t)$ must always be increasing \\[5pt]
\item $y(t)$ must always be positive
\item $y(t)$ must always be negative \\[5pt]
\item $y(t)$ must never change sign.
\end{parts}
\bookonlynewpage
\begin{annotation}
\begin{goals}
Similar to some practice problems.
Skip if the other exercises take too long.
\end{goals}
\end{annotation}
\question Consider the differential equation $2xy'=y$.
\begin{parts}
\item Check that the curves of the form $y^2 + C x = 0$ satisfy the differential equation.
\item Sketch one solution of the differential equation.
\item Sketch all the integral curves for the differential equation.
\item What is the difference between a solution passing through the point $(1,-1)$ and an integral curve passing through the same point?
\end{parts}
\standardonlynewpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% MODULE - Slope Fields
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{module}{Slope Fields}
%\Title{Slope Fields}
\label{ODE:slopefields}
\input{modules/module09-slopes.tex}
\input{modules/module09-slopes-exercises.tex}
\end{module}
\begin{lesson}
\Title{Slope Fields}
\Heading{Textbook}
\begin{itemize}
\item Module 9
\end{itemize}
\Heading{Objectives}
\begin{itemize}
\item Sketch a slope field
\item Use technology to create a slope field: WolframAlpha, Desmos, Geogebra, etc.
\item Interpret a slope field
\item Deduce properties of slope field from the ODE
\item Deduce properties of solution from slope field
\end{itemize}
\Heading{Motivation}
ODEs are often difficult to solve, so we need tools to be able to interpret their solutions and deduce properties without having to solve them.
Slope fields are such a tool.
Slope fields are not meant to be sketched by hand, so they shouldn't be asked to do that, except at the beginning to learn how they are sketched so they can understand them better.
Slope fields should be used to help interpret solutions of ODEs.
\Heading{Preparation for Class}
\begin{itemize}
\item Read textbook
\item Watch first video (second is optional)
\item Solve the core exercise \ref{slopefields:preclass}
\end{itemize}
\Heading{Tutorials and Projects}
There is no project that targets slope fields directly.
\end{lesson}
\begin{annotation}
\begin{goals}
\Goal{Pre-Class exercise}
In class, quickly solve it.
\end{goals}
\end{annotation}
\question \label{slopefields:preclass}
Consider the slope field from the first video of the module.
\begin{minipage}{.75\textwidth}
\begin{parts}
\item If $y(0)=5$, then estimate $y(-7)$.
\item If $y(0)=a$, then $y(x)>0$ for all $x>0$. For which values of $a$ is this statement true?
\end{parts}
\end{minipage}
\hfill
\begin{minipage}{125pt}
\includegraphics*[width=125pt]{images/module9-preclass.png}
\end{minipage}
\bookonlynewpage
%\newpage
\question
\begin{minipage}{.7\textwidth}
A catapult throws a projectile into the air and we track the height (in metres) of the projectile from the ground as a function $y(t)$, where $t$ is the time (in seconds) that elapsed since the object was launched from the catapult. \\
Then, the slope fields for $y(t)$ and $y'(t)$ are shown below:
\end{minipage}\hfill
\begin{minipage}{100pt}
\includegraphics*[width=100pt]{images/module9-catapult.pdf}
\end{minipage}
\setlength{\len}{200pt}
\begin{tabular}{cc}
\includegraphics*[height=\len]{images/module9-y.png}
& \includegraphics*[height=\len]{images/module9-yprime.png} \\
Slope field for $y(t)$
& Slope field for $y'(t)$
\end{tabular}
\begin{annotation}
\begin{goals}
\begin{itemize}
\item Students should think about the initial conditions.
What is a possible value for $y(0)$? What is a possible value for $y'(0)$? \\
\item What does the second slope field tell us? The equilibrium in the slope field for $y'(t)$ is called \emph{terminal velocity}.
\item Sketch a possible solution again, but for $t\in[0,30]$.
\end{itemize}
\end{goals}
\end{annotation}
\hfill {\footnotesize(These slope fields were created using WolframAlpha)} \\
\begin{parts}
\item On the slope field, sketch a \emph{possible} solution.
\item Consider the graph of $y(t)$. Does it form a parabola? Justify your answer.
\end{parts}
\newpage
%\bookonlynewpage
\begin{annotation}
\begin{goals}
\Goal{Symmetry.}
\begin{itemize}
\item The goal is not to be very accurate, but to capture the symmetry of each of these slope fields.
\item Which property of the slope field allowed you to sketch it more quickly?
\end{itemize}
\end{goals}
\end{annotation}
\question Sketch the slope field for the following differential equations.
\begin{parts}
\item $y'=x$
\begin{tikzpicture}[xscale=0.75,yscale=0.75]
\draw[thick,-{\seta}] (-5,0) -- (5.5,0) node[above] {\small $x$};
\draw[thick,-{\seta}] (0,-5) -- (0,5.5) node[left] {\small $y$};
\draw[] (1,0) node[below] {\tiny 1};
\draw[] (0,1) node[left] {\tiny 1};
\draw[step=1,lightgray,thin] (-5,-5) grid (5,5);
\end{tikzpicture}
\begin{bookonly}
\vfil
\end{bookonly}
\begin{slidesonly}
\newpage
\questionagain
\vspace{0.75cm}
\end{slidesonly}
\item $y'=y^2$
\begin{tikzpicture}[xscale=0.75,yscale=0.75]
\draw[thick,-{\seta}] (-5,0) -- (5.5,0) node[above] {\small $x$};
\draw[thick,-{\seta}] (0,-5) -- (0,5.5) node[left] {\small $y$};
\draw[] (1,0) node[below] {\tiny 1};
\draw[] (0,1) node[left] {\tiny 1};
\draw[step=1,lightgray,thin] (-5,-5) grid (5,5);
\end{tikzpicture}
\end{parts}
\newpage
%\bookonlynewpage
\begin{annotation}
\begin{goals}
Students should be able to justify their choices .
\end{goals}
\end{annotation}
\question Consider the following slope fields:
\setlength{\len}{150pt}
\begin{tabular}{ccc}
\includegraphics*[height=\len]{images/module9-graph1}
& \includegraphics*[height=\len]{images/module9-graph2}
& \includegraphics*[height=\len]{images/module9-graph3} \\
(A) & (B) & (C) \\[10pt]
\includegraphics*[height=\len]{images/module9-graph4}
& \includegraphics*[height=\len]{images/module9-graph5}
& \includegraphics*[height=\len]{images/module9-graph6} \\
(D) & (E) & (F)
\end{tabular}
\begin{notslides}
\hfill {\footnotesize(These slope fields were created using WolframAlpha)} \\
\end{notslides}
\begin{parts}
\item Which slope field(s) corresponds to a differential equation of the form
\qquad $y'=f(x)$ \qquad ?
\item Which slope field(s) corresponds to a differential equation of the form
\qquad $y'=g(y)$ \qquad ?
\item Which slope field(s) corresponds to a differential equation of the form
\qquad $y'=h(x+y)$ \qquad ?
\item Which slope field(s) corresponds to a differential equation of the form
\qquad $y'=\kappa(x-y)$ \qquad ?
\item Which slope field(s) corresponds to a differential equation of the form
\qquad $y'=1+\big( \ell(x,y) \big)^2$ \qquad ?
\item Which slope field(s) corresponds to a differential equation of the form
\qquad $y'=1-\big( m(x,y) \big)^2$ \qquad ?
\end{parts}
\standardonlynewpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Numerical Methods
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% MODULE - Numerical Methods
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{module}{Approximating Solutions}
%\Title{Slope Fields}
\label{ODE:approximation}
\input{modules/module10-approximations.tex}
\input{modules/module10-approximations-exercises.tex}
\end{module}
\begin{lesson}
\Title{Approximating Solutions}
\Heading{Textbook}
\begin{itemize}
\item Module 10
\end{itemize}
\Heading{Objectives}
\begin{itemize}
\item Know the idea of Euler's method
\item Use Euler's method
\item Be aware of the limitations of Euler's method
\item Deduce some properties of Euler's method
\end{itemize}
\Heading{Motivation}
After sketching slope fields and observing that they can be used to get an idea of the solution, the idea to use slope fields to rigorously define an approximation of the solution should come naturally.
Euler's method is just that approximation method.
This is probably one of the most important tools for an Engineer or a Physicist studying Differential equations, since the ODEs that often arise naturally from real problems are too complicated to solve rigorously and approximating the solution might be the only way.
\Heading{Preparation for Class}
\begin{itemize}
\item Read textbook
\item Watch first video (second is optional -- algebraic approach of approximating the derivative)
\item Solve the core exercise \ref{approx:preclass}
\end{itemize}
\Heading{Tutorials and Projects}
\begin{itemize}
% \item Project \ref{proj:fishery}: \fisherytitle
\item Project \ref{proj:pursuit}: \pursuittitle
\item Project \ref{proj:epidemic}: \epidemictitle
\item Project \ref{proj:lotkavolterra}: \lotkavolterratitle
\end{itemize}
\end{lesson}
%\newpage
\begin{annotation}
\begin{goals}
\Goal{Pre-class exercise}
In class, quickly solve it. \\
\url{https://www.desmos.com/calculator/ebfn5vpudr}
\begin{center}
\begin{tabular}{c|c}
$\Delta x$ & Error \\ \hline
% 10&10.4342 \\
% 5&9.9007 \\
% 4&6.7738 \\
2.5&3.4552 \\
2&2.7050 \\
1.67&2.2269 \\
1.43&1.8937 \\
1.25&1.6475 \\
1.11&1.4581 \\
1&1.3078 \\
0.5&0.6439 \\
0.25&0.3193 \\
\end{tabular}
\includegraphics*[width=100pt]{images/module10-euler-error.pdf}
\end{center}
\begin{itemize}
\item Linear with $\Delta x$: \quad $ E \approx C \Delta x$.
\end{itemize}
\end{goals}
\end{annotation}
\question \label{approx:preclass}
Consider the initial-value problem
$$
\begin{cases}
y' = -\sin(x)+\frac{y}{20} \\
y(-10)=2
\end{cases}
$$
The solution satisfies $y(10)=\frac{20\sin(10)+400\cos(10)-2e(-401-10\sin(10)+200\cos(10))}{401} \approx 6.7738406\ldots$.
%y=\frac{20\sin(x)+400\cos(x)-2e^{\frac{x+10}{20}}(-401-10\sin(10)+200\cos(10))}{401}
\begin{parts}
\item Using some software, approximate the solution at $x=10$ for different values of $\Delta x$.
\item Calculate the error between the solution and the approximation at $x=10$ for the different values of $\Delta x$.
\item Plot the error. Is it decreasing as $\Delta x$ decreases? Does it decrease linearly / quadratically / cubicly as $\Delta x$ decreases?
\end{parts}
\bookonlynewpage
\begin{annotation}
\begin{goals}
\begin{itemize}
\item The goal is to have student's recognize that the Euler approximation ``curves slower'' than the actual solution.
\item Students can explain in words why that is the case using the way the approximations are generated.\\
\item For .2 and .4:
\begin{itemize}%enumerate}[label=(\alph*)]
\item If students learned how to solve ODEs before, then fine!
\item If students didn't learn, then tell them to use WolframAlpha: \href{http://www.wolframalpha.com/input/?i=solve+y'=y-2,y(0)=3}{\tt solve y'=y-2, y(0)=3}
% \item If students didn't learn, then ask students:
% \begin{itemize}
% \item From the approximation, what kind of function does the approximation look like? (exponential)
% \item Exponentials pass through $(0,1)$, so this looks like an exponential moved up.
% \item Try $y=a e^{bt}+c$ and find $a,b,c$.
% \end{itemize}
\end{itemize}%enumerate}
\end{itemize}
\end{goals}
\end{annotation}
\question
Consider the differential equation
$$ y' = y - 2 .$$
\begin{parts}
\item Use Euler's Method to find an approximation of the solution of this differential equation that passes through the point $(0,3)$.
\item Find the solution of the differential equation with the same initial condition.
\item Use Euler's Method to find an approximation of the solution of this differential equation that passes through the point $(0,1)$.
\item Find the solution of the differential equation with the same initial condition.
\item Compare the approximations with the actual solutions. Is there a property of the Euler's Method that you can infer?
\item Explain in words why the Method satisfies that property.
\end{parts}
\bookonlynewpage
\begin{annotation}
\begin{goals}
\Goal{Only if there is time.}
\begin{itemize}
\item The question is purposefully ambiguous.
What do we mean by approximated perfectly?
\item Ex: The IVP $y'=$ sign$(t)$ (assuming sign$(0)=1$) with $y(-5)=5$ has solution $y = |t|$ and it is captured with Euler's method if $\Delta t=\frac5k$ for any $k\in\mathbb{N}$. \\
\item Once students discuss, they'll find ODE's of the form $y'= c$ for any constant.
\item Prompt them to find other types. Show them the example above only after they tried for a bit.
Then, let them revise their Conjecture. \\
\item Ex 2: $y'=f(y+t)$ with $y(0)=0$ and $f(z) = \lfloor z \rfloor$ is approximated perfectly if $\Delta t = 1$, but not if $\Delta t$ takes any other value.
\end{itemize}
\end{goals}
\end{annotation}
\question
Which differential equations will be approximated perfectly using Euler's Method?
%
%\bookonlynewpage
%
%\question
% Consider the initial-value problem
% $$
% \begin{cases}
% y' = y \\
% y(0)=1
% \end{cases}
% $$
%
% We want to study how the approximation and the solution compare.
% The solution at the point $x=1$ is $y(1)=e$.
%
% Consider $\Delta x$ and $x_k = k \Delta x$.
% Let $y_k \approx y(x_k)$ be an approximation of the solution at $x=x_k$.
%\begin{parts}
% \item What is the approximation of the solution at $x=1$? Your solution should depend on $\Delta x$.
% \item
%\end{parts}
%
%
%\begin{align*}
% y_{k+1} &=(1+\Delta x)y_k \\
% y_k &= (1+\Delta x)^k \\
% y(1) \approx y_N &= (1+\Delta x)^N = (1+\Delta x)^{\frac{1}{\Delta x}} \\
% \\
%
%\end{align*}
%\bookonlynewpage
\standardonlynewpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Modelling with ODEs
\begin{module}{Modelling with Differential Equations}
%\Title{Slope Fields}
\label{ODE:model}
\input{modules/module11-model-odes.tex}
\input{modules/module11-model-odes-exercises.tex}
\end{module}
\begin{lesson}
\Title{Modelling with Differential Equations I}
\Heading{Textbook}
\begin{itemize}
\item Module 11
\end{itemize}
\Heading{Objectives}
\begin{itemize}
\item Start modelling physical quantities
\item Follow the procedure from chapter 1 when creating a model
\end{itemize}
\Heading{Motivation}
This is one of the main goals of this course.
Students should be given the opportunity to create models in class. They should be encouraged to follow the procedure from chapter 1, as it will improve their models.
\Heading{Preparation for Class}
\begin{itemize}
\item Read textbook
\item Read the core exercise \ref{pendulum} and solve steps 1 (define problem) and 2 (create a mind map)
\end{itemize}
\Heading{Tutorials and Projects}
\begin{itemize}
\item Project \ref{proj:xray}: \xraytitle
\item Project \ref{proj:pursuit}: \pursuittitle
\item Project \ref{proj:epidemic}: \epidemictitle
\item Project \ref{proj:lotkavolterra}: \lotkavolterratitle
\end{itemize}
\end{lesson}
%\newpage
\begin{annotation}
\begin{goals}
Students will mostly likely identify the goal as finding the position of the ball $\vec{r}(t) = \big( x(t) , y(t) \big)$. That's fine!
Later, in Step 3, try to guide the students to recognize the following:
\begin{itemize}
\item Rope is massless (negligible)
\item Rope doesn't bend (negligible)
\item So can assume that the rope is rigid. How does that affect the position of the ball?
\item No friction (negligible)
\item \textbf{Important:} Students always focus on string tension. One can consider it, but it all cancels out. It's one of the exercises of the module (above). For the lecture, don't consider tension.
\end{itemize}
Then on Step 4, guide students to recognize that they actually only need to find a model for the angle, because the position of the ball really only depends on the angle $\theta(t): \vec{r}\big(\theta(t)\big)$ \\