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MATH 316 \\
ODEs \\
TR 8:00-9:15 \\   
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Instructor: J. Ellison \\
Office Hours:  TWR 9:30-10:30   
or by appointment.  \\
Office: 454 Humanities \\
Phone: 277-4110 \\
Email: ellison@math.unm.edu \\
URL: http://www.math.unm.edu/\~ \ ellison/Math316
\par\noindent  
Texts: Brannan and Boyce(BB) {\bf Differential Equations}, 
 and Polking and Arnold(PA) {\bf Ordinary Differential Equations using Matlab}, 
 third edition.\\
NOTES:
\begin{enumerate}
\item I will be away the first week of classes.  Pavel Lushnikov will lecture.
\item We are in the process of making CS151 (or equivalent experience) a prerequisite.  
Cs 151 is an introduction to Matlab and uses the book ``Matlab for Engineers'' by Holly 
Moore.  We will use Matlab in 316 and chapters 1 and 2 of PA give a good overview.
If you are not familiar with Matlab please plan to spend some extra time at the beginning of the 
course studying these two chapters or, you might consider either taking or auditing
CS151 concurrently. If you have questions, please feel free to email me.
\end{enumerate}

\begin{center} 
Course Outline 
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\begin{itemize} 
\item Weeks 1-2: First Order Equations $\frac{dy}{dx}=f(x,y)$\\
Here the emphasis is on geometry, solution techniques and numerical approximations.
  \begin{itemize}
  \item Direction fields, solutions and Euler's method
  \item Linear $y'+p(x)y=g(x)$; integrating factor and variation of parameter  
  \item Separable equations $M(x)+N(y)\frac{dy}{dx}=0$
  \item Improved Euler and variable step methods
  \item Phase line and stability for the autonomous case
  \end{itemize}    
See Sections 1.1-1.3, 2.1-2, 2.5, 2.7-8 in BB and Chapters 3 and 5 in PA\\
Optional material from other sections in Chapter 2 if time.
Students without Matlab experience should see the instructor ASAP and consider
taking or auditing CS151.

\item Weeks 3-4: Systems of First Order DEs 2x2 case\\
                  $x'=ax+by; y'=cx+dy$
  \begin{itemize}
  \item Matrix formulation and elementary matrix manipulations
  \item Eigen problem and general solution
  \item Enough theory to argue that the general solution (i.e., the set
        of all solutions) is a LC of two LI solutions
  \item Phase Plane
  \item Fundamental Solution Matrix (briefly)
%  \item Nonhomogeneous case and Variation of Parameters
  \item 4th order Runge Kutta
  \end{itemize} 
Most of Chapter 3

\item Week 5 Homogeneous Second Order Equations $ay''+by''+cy=0$
  \begin{itemize}
  \item Constant coefficient homogeneous
  \item Enough theory to argue that the general solution in the homogeneous case is 
        a linear combination of two LI solutions
  \end{itemize}
Sections 4.1-4.4

\item Week 6: Catch up and Exam

\item Week 7: Nonhomogeneous Second Order Equations $ay''+by''+cy=g(t)$
    \begin{itemize}
    \item General solution = G.S. of homogeneous plus any particular solution
    \item Method of Undertermined Coefficients
    \item Variation of Parameters - Main point it always works, but more complicated than UC
  \item Harmonic and forced harmonic motion
  \end{itemize}   
Relevant Sections 4.5-4.8

\item Weeks 8-9: Laplace Transform\\ 
Chapter 5

\item Weeks 10-11 More Systems  

\item Week 12: Catch up and Exam

\item Weeks 13-15: Nonlinear Equations and Stability
  \begin{itemize}
  \item Autonomous systems in the plane\\
        $x'=f(x,y); y'=g(x,y)$
    \begin{itemize}
    \item Equilibrium solutions and stability
    \item Linearization about equlibrium solutions
    \item Phase plane portraits with emphasis on $x''+g(x)=0$ and the 
          energy method
    \end{itemize}
  \end{itemize} 

\end{itemize} 

\pagebreak


\begin{center}
Main Goals in the context of the system $y'=f(t,y)$
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\begin{itemize}  
     \item Review of Calculus
     \item Special solution techniques 
     \item Introduction to the Laplace transform as a basic tool in Applied Mathematics as well
           as a solution technique for linear ODEs
     \item Introduction to numerical methods for integrating ODEs
       \begin{itemize}
       \item Goal: Integrate a system of ODEs using the state of the art integrators
	           in Matlab
       \end{itemize} 
      \item Intro to the linear Algebra of 2x2 matrices
        \begin{itemize}
        \item Solubility of $Ax=b$ and understanding of eigen problem $Ax=\lambda x$ 
        \end{itemize}
      \item Introduction to the qualitative theory of ODEs\\
            (Geometry of phase space)
\end{itemize}



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Grading 
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\begin{itemize}  
     \item Homework 200pts\\ 
                       Problems assigned each class time.
                       Previous weeks assignments due on Thursdays. \\
	               Home work is the most important part of the 
		       learning experience. Discussing the HW with 
		       others is okay, but the write up should 
 		       reflect your knowledge.  HWs that look too
		       much alike will not be counted.	
      \item Two midterms at 100 pts each                                     200 pts.
      \item Final: cummulative final  ~200 pts.
      \item Make-up policy: No make-up for late assignments; 
                      Solutions will be posted on Fridays.
      \item Attendance may be part of your grade. Please plan to visit office
	    hours at least once a month with a prepared question(s) so I can get 
            to know you better.

\end{itemize} 

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