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\noindent
{\bf Math 325, Spring 2001}

\begin{center}
{\bf Review for Final}
\end{center}

\noindent
{\bf Themes}

\bigskip

{\bf Note:}  Only themes 6 and 7 are new since the midterm.

\bigskip

1. Linear problems
\begin{itemize}
\item The set of solutions of a linear homogeneous problem
is a vector space.
\item Two solutions of an inhomogeneous linear problem
differ by a solution of the corresponding homogeneous
problem. 
\end{itemize}

2. An existence and uniqueness theorem tells you 
\begin{itemize}
\item there is a solution to a problem satisfying the hypotheses;
\item there is only one solution.
\end{itemize}


3.  Once you have found enough independent solutions to a
  homogeneous 
linear problem $Ly=0$,
\begin{itemize}
\item you can find all solutions;
\item you can find all solutions to $Ly=g$ starting with a 
particular solution $y_p$.
\end{itemize}

4. Good educated guesses often lead to solutions.

\medskip

5.  Transform a problem to a simple one, solve that, transform that
solution back to a solution of the original problem.

\medskip

6.  Approximate a nonlinear problem by a linear one.

\medskip


7.  Look for solutions of a simple form; try to use them to build
all solutions.

\bigskip

\noindent
{\bf Specific Topics}

\bigskip

{\bf Note:} Only topics 5-7 are new since the midterm.

\bigskip

1.  Higher order linear ODE
\begin{itemize}
\item Existence, uniqueness for initial value problem
\item Solutions of $n$th order homogeneous equation form an $n$
  dimensional
vector space 
\item Method of solving constant coefficient homogeneous equations
\end{itemize}

2. Numerical methods
\begin{itemize}
\item Euler's method, estimate for local truncation error
\item Improved Euler
\item Runge-Kutta
\item Stability
  \begin{itemize}
  \item Importance
  \item Tests, methods of judging reliability of computer output
    (controlling error in dsolve(\dots,numeric), examining graphical output)
  \end{itemize}
\end{itemize}

3. Solving ODE with Maple
\begin{itemize}
\item symbolic solution using dsolve
\item numerical solution using dsolve
\end{itemize}

4. Systems of first order linear ODE
\begin{itemize}
\item Existence, uniqueness
\item The solutions of an $n\times n$ linear homogeneous system 
form an $n$ dimensional vector space
\item Constant coefficient systems
  \begin{itemize}
  \item diagonalizable, real eigenvalues
  \item diagonalizable, complex eigenvalues
  \item not diagonalizable
    \begin{itemize}
    \item Jordan Canonical Form
    \item only did real eigenvalues in this case
    \item know how to find Jordan Canonical Form
     in $2\times 2$ case
    \item know how to use it in general case
    \end{itemize}
  \end{itemize}
\item Trajectories
  \begin{itemize}
  \item interpretation of eigendirections
  \item how to tell direction of motion
  \item behavior as $t\to\pm\infty$
  \end{itemize}
\item Vector field
  \begin{itemize}
  \item use in determining direction
  \item stability, type of critical point at origin
  \end{itemize}
\item inhomogeneous system
  \begin{itemize}
  \item Not necessarily autonomous
  \item Trajectory not necessarily independent of 
    \begin{math}
      t
    \end{math}
  \end{itemize}
\end{itemize}



5. Nonlinear systems of first order ODE
\begin{itemize}
\item Autonomous systems
  \begin{itemize}
  \item critical points, stability
  \item using the linearization to determine type, stability, when
     possible
  \end{itemize}
\item More complicated cases
  \begin{itemize}
  \item Repeated real eigenvalues (proper node, improper node or spiral point)
  \item Imaginary eigenvalues (center or spiral point, indeterminate type)
  \end{itemize}
\item Use of phase portraits, direction fields to determine type, stability
\end{itemize}



6.  PDE, Fourier series
\begin{itemize}
\item Separation of variables, especially for the heat equation
\end{itemize}
\begin{itemize}
\item Fourier series
  \begin{itemize}
  \item What they are
  \item If it looks like one, it is
  \item Convergence theorem
  \item Sine series, Cosine series
  \end{itemize}
\item More on the heat equation
\item The wave equation
\end{itemize}



7.  The Laplace transform
\begin{itemize}
\item Definition, use in solving initial value problems for ODE
\item Discontinuous forcing functions (the Heaviside function or unit
  step function)
\item Impulse forcing functions (the delta function)
\end{itemize}


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