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\centerline{Mathematics 304 -- Ordinary Differential Equations}
\centerline{Review Sheet -- Exam 2}
\centerline{November 12, 2004}
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{\it General Information}
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The second midterm exam this semester will be given in class on 
Friday, November 19, as announced in the course syllabus.  
As was true for the first exam, this will be a closed-book
test.  You may use calculators at any point.  
The exam will cover the material we have covered since the first exam
(see topics below), through topics from class on Friday, November 12.   
If there is interest, I would be happy to arrange a late afternoon
or evening review session for the exam.  Wednesday, November 17 is probably
the best day to aim for.
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\noindent
{\it Topics To Be Included}
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\item{1)}  First order systems $X' = AX$.  Classification of 
equilibrium points via the eigenvalues of $A$ and via the 
``trace-determinant plane'' in the $2\times 2$ case.  Problems 
may ask you to work with $2\times 2$, $3\times 3$, or possibly 
$4\times 4$ matrices.   Finding the general solution via the 
corresponding canonical form system $Y' = BY$ and change of basis.  
\item{2)}  Application to homogeneous harmonic oscillator equations
$mx'' + bx' + kx = 0$.  Know how to convert to a first order system
and apply 1 (with as $2\times 2$ matrix) to find the general solution.
The undamped, underdamped, critically damped, overdamped
cases, and how solutions in each are different.
\item{3)}  Forced oscillator equations $mx'' + bx' + kx = g(t)$.
The general solution is $x(t) = c_1 x_1(t) + c_2 x_2(t) + x_p(t)$
where $c_1 x_1(t) + c_2 x_2(t)$ is the general solution of the 
homogeneous equation $mx'' + bx' + kx = 0$, and $x_p(t)$ is a particular
solution.  Finding particular solutions by the method of 
undetermined coefficients for exponential, polynomial, and sinusoidal
forcing terms (including the case where the forcing
term is a solution of the homogeneous equation).  The phenomenon of 
resonance in undamped systems.
\item{4)}  Equilibrium points and linearization for nonlinear systems.
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{\it Suggested Review/Practice Problems}
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\noindent
From Hirsch, Smale, and Devaney:
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\leftline{Chapter 2/2,3}
\leftline{Chapter 3/2ii, iv, vi, 4, 5, 11, 12, 13, 14, 16 (16 would be
an ``extra credit''-type problem)}
\leftline{Chapter 4/1,2,3}
\leftline{Chapter 5/5dfgh,6}
\leftline{Chapter 6/1dfgh (note--these are the same matrices as in 5dfgh 
as above)}
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\noindent
From Blanchard, Hall, and Devaney:
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\leftline{Section 4.1/9,25,31,36,41}
\leftline{Section 4.2/1,11}
\leftline{Section 5.1/1,3,4,5 (5 is a good review
material from the first part of the course too!)}
\bye