\documentclass{article}

\setlength{\topmargin}{-.3in}
\setlength{\oddsidemargin}{-.2in}
\setlength{\textheight}{7.5in}
\setlength{\textwidth}{6.5in}



\usepackage{amssymb}


\begin{document}

\textbf{2250 - Reveiw for test 2}

\textbf{Topics:}

\begin{enumerate}

\item  Test is Monday, July 22.
\item  Test will cover Chapters 5-7.2 (Everything except 1.6, 3.7)
\item  Test will be closed book, closed notes.  
  You can use your calculator, but I expect to be able to follow your work as if you had no calculator.
\item  Linear Differential Equations - Theory
  \begin{enumerate}
  \item  Superposition for homogeneous equations.
  \item  Linear independant solutions (why is this important?)
  \item  How to tell if solutions are linearly independent.
  \item  Existance and uniqueness theorem (what does it say?)
  \item  Complimentary solutions and particular solutions: What does a general solution look like?
  \end{enumerate}
\item  Linear differential equations: solving (with constant coefficients)
  \begin{enumerate}
  \item  Characteristic equation.
  \item  Finding the complimentary solution (solving the homogeneous equation).
  \item  Finding the particular solution: Undetermined coeffients (don't worry about variation of parameters)
  \end{enumerate}
\item  Linear equations: applications
  \begin{enumerate}
  \item  Spring constant, damping constant: how are these related to the linear equation?
  \item  Forced oscilations and resonance: when is there resonance (at what forcing frequency?) and practical resonance?
  \end{enumerate}
\item  Eigenvalues 
  \begin{enumerate}
  \item  What is an eigenvalue and eigenvector?
  \item  How to find eigenvalues and eigenvectors for a matrix.
  \item  What is an eigenspace?
  \item  How to diagonalize a matrix.  What is it possible?  What does this mean?
  \end{enumerate}
\item  Linear systems of differential equations
  \begin{enumerate}
  \item  How to turn a higher order equation into a system.
  \item  Existance and uniqueness theorem for linear systems.
  \item  Linear independent solutions.
  \item  What is a general solutions to a linear system?
  \end{enumerate}
\end{enumerate}

\newpage
\textbf{Some things you should make sure you can do (``sample test,'' not an exhaustive list)}

\begin{enumerate}
\item  Solve a linear equation with constant coefficients (find a general solution to homogeneous or nonhomogeneous equation).
\item  Given an equation, know and apply the existance and uniqueness theorem.
\item  Given a damped spring situation, set up (and solve) the differential equation.
\item  Given a matrix, find all eigenvalues and a basis for each eigenspace.
\item  Given a matrix, determine if the matrix is diagonalizable.  If diagonalizable, diagonalize the matrix.
\item  Turn a higher order system into a linear system.
\item  Check to see if a vector of functions is a solution to a linear system.
\item  Check if a set of vectors of functions are linearly independent.
\item  Write a linear system as a matrix differential equation.
\end{enumerate}






\end{document}







