chap2-firstorder.tex

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%		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 $\theta(t): \vec{r}\big(\theta(t)\big)$ \\

		Students should finish it at home.				
	\end{goals}
\end{annotation}

\question \label{pendulum}
	A pendulum is swinging side to side. We want to model its movement.

\begin{minipage}{.7\textwidth}
\begin{parts}
	\item Define the problem. Which function(s) do we want to find in the end?
	\item Build a mind map.
	\item Make assumptions. Remember to use your mind map to help structure the problem.
	\item Construct a model. You should end up with one (or more) differential equations. 
	
	Remember that there are some Physics principles that can help you (e.g. Newton's 2$^{\rm nd}$ Law, Conservation of Energy, Linear Momentum, and Angular Momentum, Rate of Change is Rate in $-$ Rate out).
	\item Assess your model:
	\begin{enumerate}
		\item Find one test that your model passes.
		\item Find one test that your model fails.
	\end{enumerate}
	
\end{parts}
\end{minipage}\hfill
\begin{minipage}{100pt}
	\includegraphics*[width=100pt]{images/module11-pendulum.pdf}	
\end{minipage}









\bookonlynewpage

\begin{notslides}
\hfill


\bookonlynewpage
\end{notslides}

\begin{lesson}
	\Title{Modelling with Differential Equations II}

\Heading{Textbook}
	\begin{itemize}
		\item Module 11
	\end{itemize}

\Heading{Objectives}
	\begin{itemize}
		\item Model an open ended problem
		\item Follow the procedure from chapter 1 when creating a model
	\end{itemize}
	
\Heading{Motivation} 

Students may be used to some models from physics classes or from Calculus classes. 

They should practice developing more open ended models.

\Heading{Preparation for Class}
\begin{itemize}
	\item Read textbook
	\item Read the core exercise \ref{rumour} and solve steps \hyperref[moddefine]{1} (define problem) and \hyperref[mindmap]{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}


\question \label{rumour} Model the spreading of a rumour through the students of a school.

\textbf{Hint.} Start with a simple model and then include more details.



\standardonlynewpage

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%		Solvable Types of ODEs: Separable, First-Order Linear, Exact (?)


\begin{module}{Solvable Types of ODEs}
	%\Title{Separable ODEs}

	\label{ODE:solve}

	\input{modules/module12-SolvableODEs.tex}
	\input{modules/module12-SolvableODEs-exercises.tex}

\end{module}


\begin{lesson}
	\Title{Solvable Types of Differential Equations}

\Heading{Textbook}
	\begin{itemize}
		\item Module 12
	\end{itemize}

\Heading{Objectives}
	\begin{itemize}
		\item Identify a Separable ODE
		\item Know how to solve a Separable ODE
		\item Identify a first-order linear ODE
		\item Know how to solve a first-order linear ODE: Method of the Integrating Factor
	\end{itemize}
	
\Heading{Motivation} 

This is the traditional class on differential equations. 

Even though the focus of the course is not on recipes for solving differential equations, when creating a model that involves a differential equation and analyzing it, it is necessary to be aware of some of the basic methods for solving them. 

Students should practice these methods more than the limited time devoted to them in lecture. Even though there aren't many practice problems, it is easy to find exercises for this module in any differential equations textbook.

\Heading{Preparation for Class}
\begin{itemize}
	\item Read textbook
	\item Watch first video for each method
\end{itemize}


\Heading{Tutorials and Projects}
\begin{itemize}
	\item Project \ref{proj:fishery}: \fisherytitle
	\item Project \ref{proj:xray}: \xraytitle
	\item Project \ref{proj:pursuit}: \pursuittitle
\end{itemize}


\end{lesson}




%\newpage


\begin{annotation}
\begin{goals}
	\begin{itemize}
		\item First, ask students to identify all the ODEs
		\item Then, students can start solving them one by one \\
	\end{itemize}
\end{goals}	
\begin{notes}
	\begin{itemize}
		\item Make sure there is time for next core exercise. Skip some ODEs if necessary.
	\end{itemize}	
\end{notes}
\end{annotation}

\question 
	Decide whether the following differential equations are separable, first-order linear, both, or neither. If they are of one of the solvable types, solve it.

\begin{parts}
	\item $\displaystyle\theta''(t) = \frac{g}{L} \sin\big(\theta(t)\big)$
	\item $\displaystyle P'(t) = r P(t) \left( 1 - \frac{P(t)}{K} \right)$
	\item $\displaystyle v'(t) = -g - \frac{\gamma}{m} v(t)$
	\item $\displaystyle y'(t) = -gt - \frac{g}{m} y(t) + 10$
\end{parts}











\bookonlynewpage




\begin{annotation}
\begin{goals}
	\Goal{Idea of Integrating Factor}
	By recognizing that the left-hand side is the result of a product rule, it is much easier to find the solution.
\end{goals}
\end{annotation}
\question
\begin{parts}
	\item Calculate $\big( \sin(x) f(x) \big)'$.
	\item Find the general solution of 
		$$ \sin(x) y' + \cos(x) y = \sqrt{x}.$$
	\item What is the integrating factor for the differential equation
		$$ y' + \frac{\cos(x)}{\sin(x)} y = \frac{\sqrt{x}}{\sin(x)}$$
\end{parts}









%
%\bookonlynewpage
%
%\hfill

\standardonlynewpage



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%		Properties of ODEs


\begin{module}{Properties of Differential Equations}
	\label{ODE:properties}
	%\Title{First-Order Linear ODEs}
%	\Heading{Textbook}	
%	\Heading{Objectives}
%	\begin{itemize}
%		\item Bla bla bla	
%	\end{itemize}

	\input{modules/module13-PropertiesODEs.tex}
	\input{modules/module13-PropertiesODEs-exercises.tex}



%	\Heading{Motivation} 



\end{module}



\begin{lesson}
	\Title{Properties of Differential Equations}

\Heading{Textbook}
	\begin{itemize}
		\item Module 13
	\end{itemize}

\Heading{Objectives}
	\begin{itemize}
		\item Identify which Theorem to use for a given ODE
		\item Know the Existence and Uniqueness Theorems
		\item Deduce properties of the solutions from the Theorems
	\end{itemize}
	
\Heading{Motivation} 

This is a little glimpse into the theory of Differential Equations. 

Knowing whether a differential equation has solutions, and whether there is one unique solution is important on its own right.

But even from a more applied point-of-view, these Theorems offer consequences to the behaviour of solutions of differential equations that make them very important. 

One of these properties is the fact that in the region in the $(t,y)$-plane where the Theorems' conditions are satisfied, solutions cannot intersect.


\Heading{Preparation for Class}
\begin{itemize}
	\item Read textbook
	\item Watch first video: about the Theorem for Nonlinear ODEs
	\item Watch the second video: short example video
	\item Solve the core exercise \ref{prop:preclass}
\end{itemize}


\Heading{Tutorials and Projects}
\begin{itemize}
	\item Project \ref{proj:fishery}: \fisherytitle
	\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}
	\Goal{Pre-Class exercise}	
	In class, quickly solve it.
\end{goals}	
\end{annotation}
\question \label{prop:preclass}Consider the example in the video:
$$
\begin{cases}
x \dfrac{dy}{dx} = y \\
y(0)=b
\end{cases}
$$

Without solving, but only according to the Existence and Uniqueness Theorem, what can we conclude?
\begin{enumerate}[label=\color{gray}(\alph*)]
	\item We can conclude that there is a unique solution.
	\item We can conclude that if $b=0$ there are many solutions, but if $b\neq 0$, then there are no solutions.
	\item We can conclude that there are many solutions.
	\item We can conclude that there are no solutions.
	\item We can't conclude anything.
\end{enumerate}




\bookonlynewpage

\question For the following initial-value problems, answer the following questions:
\begin{enumerate}[label=\color{gray}(\alph*)]
\item Is there a unique solution?

\item Without solving, what is its domain?
\end{enumerate}
\begin{parts}
	\item $y' = t + \frac{y}{t-\pi}$ with $y(1) = 1$
	\item $y' = t + \sqrt{y-\pi}$ with $y(1) = 1$
	\item $y' = \sqrt{4 - (t^2+y^2)}$ with $y(1) = 1$
\end{parts}






\bookonlynewpage

\begin{minipage}{.5\textwidth}
\question Consider a differential equation $y' = f(t,y)$ where 
\begin{itemize}
	\item $f(t,y)$ is continuous for all $t,y \in \mathbb{R}$;
	\item $\frac{\partial f}{\partial y}(t,y)$ is continuous for all $t \in \mathbb{R}$,$y>0$.
\end{itemize}
\end{minipage}
\hfill
\begin{minipage}{250pt}
%Consider the following graph:
%\begin{center}
\begin{tikzpicture}[scale=0.9]
  \draw[thick,-{\seta}] (-4,0) -- (4,0) node[above] {\small $t$};
  \draw[thick,-{\seta}] (0,-2) -- (0,2) node[left] {\small $y$};
  \draw[samples=40, ultra thick,domain=-4:4,smooth,variable=\x,cyan] plot ({\x},{sin(1.5*deg(\x))/2-\x/4}) node[below] {$y_3$};
  \draw[samples=40, ultra thick,domain=0.075:4,smooth,variable=\x,gray] plot ({\x},{-1/((\x^(0.33))+\x)}) node[below] {$y_2$};
  \draw[samples=40, ultra thick,domain=-2.35:3,smooth,variable=\x,green] plot ({\x-1.5},{\x*\x*\x/8-\x*\x/3+1.6}) node[right] {$y_1$};
  \draw[dashed] (1,-2) -- (1,2);
  \draw (1,0) node[below] {$t_0$};
  \draw (1,-0.5) node {\tikzcircle[gray,fill=gray!50!white]{2pt}};
  \draw (1,{(2.5^3)/8-(2.5^2)/3+1.6}) node {\tikzcircle[green,fill=green!50!white]{2pt}};
  \draw (1,{sin(1.5*deg(1))/2-1/4}) node {\tikzcircle[cyan,fill=cyan!50!white]{2pt}};
\end{tikzpicture}
%	\includegraphics*[width=250pt]{images/module13-3graphs.pdf}
%\end{center}
\end{minipage}

\begin{annotation}
\begin{goals}
	\Goal{Intersecting solutions?}
	The goal is for students to understand why solutions cannot touch \emph{IF} the Theorem is valid at the intersection point.
	
	\begin{itemize}
		\item  $y1$, $y_3$ not ok!
		\item  $y1$, $y_2$ ok!
		\item  $y2$, $y_3$ ok!
	\end{itemize}
\end{goals}	
\end{annotation}
\begin{parts}
	\item Can \textbf{\color{green} green $y1$} and \textbf{\color{cyan} blue $y3$} be two solutions of the same differential equation above with two different initial conditions? Why?
	\item Can \textbf{\color{green} green $y1$} and \textbf{\color{gray} gray $y2$} be two solutions of the same differential equation above with two different initial conditions? Why?
	\item Can \textbf{\color{gray} gray $y2$} and \textbf{\color{cyan} blue $y3$} be two solutions of the same differential equation above with two different initial conditions? Why?

	\item Based on the answers to the three parts above, write a Corollary to the Existence and Uniqueness Theorems.
\end{parts}









\bookonlynewpage


\begin{annotation}
\begin{goals}
	\Goal{Existence and Uniqueness Thms}
	The goal is for the students to realize that if the Theorem's conditions hold, then its result must also hold, even though it might not look like it.
	
	So if the Theorem says that the solution is unique and we see two solutions, then they must be the same function (at least for $x$ near $\frac12$).
\end{goals}
\end{annotation}
\question The initial-value problem
$$
\begin{cases}
y' = -\dfrac{x}{y} \\
y\big(\frac12\big) = \frac{\sqrt{3}}{2}.
\end{cases}
$$
has the solutions

\hfil $y_1(x) = \cos\big( \arcsin (x) \big) \qquad \text{ and } \qquad y_2(x) = \sqrt{1-x^2}$ \quad . 

\begin{parts}
\item Does the problem satisfy the conditions of one of the Existence and Uniqueness Theorems?

\item What can you conclude?

\end{parts}

\paragraph{Hint.} ``When you have eliminated the impossible, whatever remains, however improbable, must be the truth.'' -- Sherlock Holmes




%\bookonlynewpage

\standardonlynewpage



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%		Autonomous of ODEs


\begin{module}{Autonomous Differential Equations}
	%\Title{First-Order Linear ODEs}
%	\Heading{Textbook}	
%	\Heading{Objectives}
%	\begin{itemize}
%		\item Bla bla bla	
%	\end{itemize}

	\label{ODE:autonomous}

	\input{modules/module14-AutonomousODEs.tex}
	\input{modules/module14-AutonomousODEs-exercises.tex}



%	\Heading{Motivation} 



\end{module}



\begin{lesson}
	\Title{Autonomous Differential Equations}

\Heading{Textbook}
	\begin{itemize}
		\item Module 14
	\end{itemize}

\Heading{Objectives}
	\begin{itemize}
		\item Identify an Autonomous ODE
		\item Find and plot equilibrium solutions
		\item Sketch solutions of an autonomous ODE without solving it
		\item Classify equilibrium solutions
	\end{itemize}
	
\Heading{Motivation} 

Even though nonlinear ODEs are in general very hard to study, Autonomous ODEs, which are often nonlinear, are an exception. 

These are ODEs of the form $y'=f(y)$, which can usually be thoroughly studied without having to find their solution. 

One of the key ways that we use to study Autonomous equations is to find equilibrium points and classify them as stable / unstable.
This will be extended to systems of ODEs.


\Heading{Preparation for Class}
\begin{itemize}
	\item Read textbook
	\item Watch the video
	\item Solve the core exercise \ref{autonomous:preclass}
\end{itemize}

\Heading{Tutorials and Projects}
\begin{itemize}
	\item Project \ref{proj:fishery}: \fisherytitle
	\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}


\begin{annotation}
\begin{goals}
	\Goal{Pre-Class exercise}
	In class, quickly solve it.
\end{goals}	
\end{annotation}
\question \label{autonomous:preclass}
	Consider the ODE from the video in the module.

\begin{center}
	\includegraphics*[width=190pt]{images/module14-preclass.png}
\end{center}
Select all the initial conditions that yield a decreasing solution.
\begin{enumerate}[label=\color{gray}(\alph*)]
	\item $x(-2)=\sqrt{2}$
	\item $x(20\,000)=0.000000001$
	\item $x(5)=\pi$
	\item $x(0)=\frac12$
	\item $x(3)=-\frac12$
	\item $x(1000) = \frac1e$
\end{enumerate}




\bookonlynewpage


%\newpage

\begin{annotation}
\begin{goals}
	\begin{itemize}
		\item \emph{1--3}: Students come up with a definition of stable, unstable, semi-stable equilibrium points.
		\item \emph{4}: Sketch solutions (do (a),(b) in class)
		\item \emph{5--6}: Find properties of solutions
		\item \emph{7}: Leave as HW
	\end{itemize}
\end{goals}	
\end{annotation}
\question Consider the differential equation \quad $y'=f(y)$ \quad where $f(y)$ is given by the following graph:
\begin{center}
	\begin{tikzpicture}
	  \draw[thick,-{\seta}] (-4.5,0) -- (6,0) node[above] {\large $y$};
	  \draw[thick,-{\seta}] (0,-3) -- (0,3) node[left] {\large $y'$};
	  \foreach \k in {-4,...,-1} {
	    \draw ({\k},0) node[above] {$\k$};
	    \draw[dashed,color=gray] ({\k},-3) -- ({\k},3);
	  }
	  \foreach \k in {1,...,5} {
	    \draw ({\k},0) node[below] {$\k$};
	    \draw[dashed,color=gray] ({\k},-3) -- ({\k},3);
	  }
	  \foreach \k in {-2,-1,1,2} {
	    \draw (0,{\k}) node[left] {$\k$};
	    \draw[dashed,color=gray] (-4.5,{\k}) -- (6,{\k});
	  }
	  \draw[samples=10, thick,domain=-1.5:0,smooth,variable=\x,cyan] plot ({\x-3},{-2*exp(2.4*\x)});
	  \draw[samples=10, thick,domain=-1:0,smooth,variable=\x,cyan] plot ({\x},{\x});
	  \draw (-1,-1) node {\tikzcircle[thick,cyan,fill=white]{2pt}};
	  \draw (-1,0) node {\tikzcircle[thick,cyan,fill=cyan]{2pt}};
	  \draw[samples=10, thick,domain=-3:-1,smooth,variable=\x,cyan] plot ({\x},{\x+1});
	  \draw[samples=10, thick,domain=0:1,smooth,variable=\x,cyan] plot ({\x},{sqrt(\x)});
	  \draw[samples=10, thick,domain=-1:1,smooth,variable=\x,cyan] plot ({\x+2},{\x*\x});
	  \draw[samples=20, thick,domain=0:3,smooth,variable=\x,cyan] plot ({\x+3},{sin(deg(pi*\x))/2+1+\x/2});
	\end{tikzpicture}
	%\includegraphics*[width=350pt]{images/module14-graphf.pdf}
\end{center}

\begin{parts}
	\item What are the equilibrium points?
	\item Which equilibrium solutions are stable, unstable, or semi-stable?
	\item Write a definition for a \textbf{stable}, \textbf{unstable}, and \textbf{semi-stable} equilibrium point.
	\item Roughly, sketch a solution satisfying:
	\begin{enumerate}
		\item $y(0)=2.5$.
		\item $y(0)=-\frac14$.
		\item $y(1)=\frac14$.
	\end{enumerate}
	\item If $y(0) = 2$, then $y(t) = $
	\item If $y(0) = \frac12$, then $\displaystyle \lim_{t\to \infty} y(t) = $
	\item If $y(0) = -2$, then $\displaystyle \max_{t \in [0,\infty)} y(t) = $
\end{parts}


\bookonlynewpage

\begin{notslides}


\hfill


\bookonlynewpage
	
\end{notslides}


\question Consider a differential equation $y' = f(t,y)$ with the following slope field.

\begin{center}
	\def\delta{0.25}
	\def\NN{16}% \NN = 4/\delta
	\def\fac{0.75}
	\begin{tikzpicture}[scale=1.5]
	  \draw[thick,-latex] (-0.5,0) -- (4.5,0) node[above] {\large $t$};
	  \draw[thick,-latex] (0,-1.5) -- (0,3) node[left] {\large $y$};
	  \foreach \k in {-1,...,2} {
	    \draw (0,{\k}) node[left] {$\k$};
	    \draw (-0.05,{\k})--(0.05,{\k});
	  }
	  \foreach \xk in {0,...,\NN} {
	    \foreach \yk in {0,...,\NN} {
	      \draw[-latex,cyan] ({\xk*\delta},{(-1.5+\yk*\delta)}) -- ({\xk*\delta+(0.5+abs(cos(deg(pi*(-1.5+\yk*\delta)))))*\delta*\fac},{(-1.5+\yk*\delta)+abs(sin(deg(pi*(-1.5+\yk*\delta))))*\delta*\fac});
	    }
	  }
	\end{tikzpicture}
%	\includegraphics*[width=200pt]{images/module12-dirfield.pdf}
\end{center}

\begin{parts}
	\item What are the equilibrium solutions of the ODE?


\item Directly on the direction field above, sketch the solution of the problem
$$
\begin{cases}
y'=f(t,y) \\
y(0) = \dfrac14
\end{cases}
$$

\item From the direction field above, circle the correct type(s) of this ODE? Justify your answer.

\begin{enumerate}[label=\color{gray}(\alph*)]
\hfil 
\begin{minipage}{.3\textwidth}
\item separable. \\[-10pt]
\item of first-order and linear.
\end{minipage}
\hfil
\begin{minipage}{.4\textwidth}
\item autonomous. \\[-10pt]
\item none of the other options.
\end{minipage}
\end{enumerate}

\item Assume that $y=g(t)$ and $y=h(t)$ are two solutions of the differential equation with $g(0)<h(0)$, then

\hfill (select all the possible options)\\[-10pt]

\begin{enumerate}[label=\color{gray}(\alph*)]
\hfil 
\begin{minipage}{.2\textwidth}
\item $g(3) < h(3)$
\end{minipage}
\hfil
\begin{minipage}{.2\textwidth}
\item $g(3) = h(3)$
\end{minipage}
\hfil
\begin{minipage}{.2\textwidth}
\item $g(3) > h(3)$
\end{minipage}
\end{enumerate}

\end{parts}




\bookonlynewpage

\standardonlynewpage

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%%		Analyzing ODEs
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%\begin{module}{Analysis of Differential Equations}
%	%\Title{First-Order Linear ODEs}
%%	\Heading{Textbook}	
%%	\Heading{Objectives}
%%	\begin{itemize}
%%		\item Bla bla bla	
%%	\end{itemize}
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%	\label{odes-analysis}
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%	\input{modules/module15-AnalysisODEs.tex}
%	\input{modules/module15-AnalysisODEs-exercises.tex}
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%%	\Heading{Motivation} 
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%\end{module}
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%\begin{lesson}
%	\Title{Analysis of Differential Equations}
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%%	\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}
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%\end{lesson}
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%\newpage
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%\question 
%	Bla!
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