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\lhead{Your name}
\rhead{Student ID}
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\item[\hskip \labelsep {\bfseries #1}\hskip \labelsep {\bfseries #2.}]}{\end{trivlist}}

\begin{document}

\date{}
\title{Problem Set 4 - Due November 18, at 11:59pm}
\author{[Your name] and [student ID]\\
MATB44H3-2019}
\maketitle
 
\begin{problem}{1}
Let $\dot{x} = Ax(t)$, with $x(0) = x_0$, where $A$ is a 2x2 matrix. Plot the phase portrait for each of the following matrices:
\begin{enumerate}
    \item [(i)] 
$
A=
\begin{bmatrix}
    -1 & 2 \\
    2 & 2
\end{bmatrix}
$
    \item [(i)] 
$
A=
\begin{bmatrix}
    -2 & 2 \\
    -1 & 0
\end{bmatrix}
$
    \item [(i)] 
$
A=
\begin{bmatrix}
    -1 & 0 \\
    -3 & -4
\end{bmatrix}
$
    \item [(i)] 
$
A=
\begin{bmatrix}
    -3 & 4 \\
    -2 & 3
\end{bmatrix}
$

\end{enumerate}


\end{problem}
\begin{proof}[Solution]
Collaborators: Donald Duck and Mickey Mouse
\end{proof}
 
\newpage
 
\begin{problem}{2 (Teschl 3.9)}
Let $\dot{x} = Ax(t)$, with $x(0) = x_0$. Solve the systems corresponding to the following matrices:
\begin{enumerate}
    \item [(i)] 
$
A=
\begin{bmatrix}
    2 & 1 \\
    0 & 2
\end{bmatrix},
\qquad
x_0 =
\begin{bmatrix}
    1 \\
    1
\end{bmatrix}
$
    \item [(i)] 
$
A=
\begin{bmatrix}
    -1 & 1 \\
    0 & 1
\end{bmatrix},
\qquad
x_0 =
\begin{bmatrix}
    1 \\
    -1
\end{bmatrix}
$
\end{enumerate}
\end{problem}

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

\newpage
 
\begin{problem}{3 (Teschl 3.10)}
Solve
\[
\dot{x} = -y - t, \qquad \dot{y} = x + t, \qquad x(0)=1, y(0)=0
\]
\end{problem}


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


\newpage

\begin{problem}{4 (Teschl 3.14)}
Let $A$ by a real 2-by-2 matrix. Then the eigenvalues can be expressed in terms of the determinant $D = \det(A)$ and the trace $T = \textrm{tr}(A)$.
In particular, $(T,D)$ can take all possible values in $\mathbb{R}^2$ if $A$ ranges over all possible matrices in $\mathbb{R}^{2 \times 2}$. Split the $(T, D)$ plane into regions in which the various cases discussed in lecture / Teschl Section 3.2 occur (source, spiral source, sink, spiral sink, saddle, center).
\end{problem}

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


\end{document}
