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\lhead{Your name}
\rhead{Student ID}
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\begin{document}
 
\title{Problem Set 2 - Due Oct 7, 11:59pm}
\author{[Your name] and [student ID]\\
MATB44H3-2019}
\maketitle
 
\begin{problem}{1}
Consider the autonomous equation 
\begin{equation*}
    \frac{dx}{dt} = - \left( 1 - \frac{x}{T} \right) \left( 1 - \frac{x}{K} \right) x ,
\end{equation*}
for constants $T, K \in \mathbb{R}$ and $0 < T < K$.
When $x>0$, this equation can be thought to describe logistic growth with a quorum.
We wish to qualitatively analyze the limiting behavior of this system as $t \to \infty$.
\begin{itemize}
    \item Plot a graph of $\frac{dx}{dt}$ (vertical) vs. $x$ (horizontal).
    \item Draw the direction field along the $x$ axis, with arrows pointing the direction the system will evolve.
    \item The zeros of $\frac{dx}{dt}$ are called its \textbf{critical points}. How many critical points are there? What are they, and which ones are stable?
\end{itemize}

\end{problem}
\begin{proof}[Solution]
Collaborators: Donald Duck and Mickey Mouse
\end{proof}
 
 
\begin{problem}{2}
Find the first $3$ Picard iterations after $x_0(t)$ of the particular solution of each of the following equations:
\begin{enumerate}
\item[(i)] $\dot{x} = t + x$, where $x(0) = 1$
\item[(ii)] $\dot{x} = e^t + x$, where $x(0) = 0$
\end{enumerate}
\end{problem}

\begin{proof}[Solution]
Collaborators: Gandalf Greyhame and Bilbo Baggins
\end{proof}

 
\begin{problem}{3}
Given a function $x: \mathbb{R} \to \mathbb{R}$, prove that the map
\begin{equation*}
    ||x|| \myeq \sup_{t \in [0,1]} |x(t)|
\end{equation*}
is a norm.
\end{problem}


\begin{proof}[Solution]
Collaborators: Mark Watney
\end{proof}



\begin{problem}{4}
Recall Newton's method for finding the zeros of a twice continuously differentiable function $f(x)$, given by
\begin{equation*}
    x_{n+1} = K(x_n), \vspace{1em} K(x) = x - \frac{f(x)}{f'(x)}.
\end{equation*}
That is to say, we start from a point $x_0$, and then repeatedly apply $K(x)$ to approximate a nearby $0$ of $f(x)$ through the sequence $x_0, x_1, \ldots$.

Prove that Newton's method works:
\begin{itemize}
    \item First show that if $f(\bar{x})=0$ and $f'(\bar{x}) \ne 0$, then $\bar{x}$ is a fixed point of $K$.
    \item Then show that there is a corresponding closed interval $C$ around $\bar{x}$ such that we can apply the Banach fixed point theorem to prove uniqueness of the solution in that interval.
\end{itemize}
\end{problem}

\begin{proof}[Solution]
Collaborators: No one.
\end{proof}


\end{document}
