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\lhead{Your name}
\rhead{Student ID}
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\begin{document}
 
\title{Problem Set 1}
\author{[Your name] and [student ID]\\
MATB44H3-2019}
\maketitle
 
\begin{problem}{(Teschl) 1.3}
Classify the following differential equations. Is the equation linear, autonomous, or homogeneous? What is its order? Note also that these classifications only apply to differential equations, so a valid answer could be that the equation is not an ODE.

\begin{enumerate}
\item[(i)] $y'(x) + y(x) = 0$
\item[(ii)] $\frac{d^2}{dt^2}u(t) = t \sin \left(u(t)\right)$
\item[(iii)] $y(t)^2 + 2y(t) = 0$
\item[(v)] $\dot{x}=-y$, $\dot{y}=x$
\end{enumerate}
\end{problem}

\begin{proof}[Solution]
\textit{Collaborators: [Collaborator 1], [Collaborator 2], [Collaborator 3]}

Write a solution here


\end{proof}

\begin{problem}{(Teschl) 1.4}
Which of the following differential equations for $y(x)$ are linear?
\begin{enumerate}
\item[(i)] $y' = \sin(x)y + \cos(y)$
\item[(ii)] $y' = \sin(y)x + \cos(x)$
\item[(iii)] $y' = \sin(x)y + \cos(x)$
\end{enumerate}
\end{problem}

\begin{proof}[Solution]
\textit{No collaborators}

Write a solution here
\end{proof}


\newpage

\begin{problem}{(Teschl) 1.5}
Find the most general form of a second-order linear equation
\end{problem}
\begin{proof}[Solution]
Write a solution here.
\end{proof}

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\begin{itemize}
\item Teschl Problem 1.3. Parts (i), (ii), (iii), (v). Make sure to say if the equation is linear, autonomous, or homogeneous, and give its order. Note that these classifications only apply to differential equations, so a valid answer is that the equation is not an ODE.
\item Teschl Problem 1.4
\item Teschl Problem 1.5
\item Teschl Problem 1.6. Prove your answer to the question about whether linearity is always preserved.
\item Teschl Problem 1.7. Explain why linearity is or is not preserved in this particular problem.
\item Teschl Problem 1.8.
\item Teschl Problem 1.9. Parts (i), (ii) only.
\item Teschl Problem 1.12
\item Teschl Problem 1.14
\item Teschl Problem 1.15
\item Teschl Problem 1.18. Parts (i), (ii) only.
\item Teschl Problem 1.22
\item Teschl Problem 1.23
\end{itemize}

\textbf{Optional bonus problems:}
\begin{itemize}
\item Teschl Problem 1.10 (bonus)
\item Teschl Problem 1.13 (bonus)
\item Teschl Problem 1.24 (bonus)
\end{itemize}


\end{document}
