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\centerline{Mathematics 304, section 2 -- Ordinary Differential Equations}
\centerline{Midterm Exam 2}
\centerline{November 19, 2004}
\noindent
{\it Directions}
\bigskip
Read each question carefully before starting to work.
Do all work in the blue exam booklet.  There are 100 regular points
and 10 Extra Credit points (see back).
\bigskip
\noindent
I.  All parts of this question refer to the family of 1st order systems
$$X' = \pmatrix{-1 & a\cr -1 & 0\cr}X\leqno(1)$$
depending on the parameter $a$.
\item{A)}  (5) What type of equilibrium point does the system in
this family with $a = 10$ have at the origin? (You 
{\it do not} need to find the general solution for this one!)
\item{B)}  (20) Find the general solution of the system 
in the family with $a = -2$.  What type of equilibrium
point does this system have at the origin?
\item{C)}  (5) Sketch the motion in the trace-determinant plane
corresponding to the family (1) as $a$ ranges over all real values.
Besides the cases from parts A and B, what other types
of phase portraits are possible in this family?
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\noindent
II.  
\item{A)} (10) Show that if $x_{p,1}$ is a particular solution of 
$mx'' + bx' + kx = g_1(t)$
and $x_{p,2}$ is a particular solution of 
$mx'' + bx' + kx = g_2(t)$, then the sum $x_{p,1}(t) + x_{p,2}(t)$
solves $mx'' + bx' + kx = g_1(t) + g_2(t)$.
\item{B)} (20) Consider the following initial value problem:
$x'' + 4x' + 3x = e^{7t} + 4e^{-t}$,
with $x(0) = 0, x'(0) = 1$.
Find the solution using part A and the method of undetermined 
coefficients.
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\noindent
III.  (15)  For what value of $\omega$ will {\it resonance} occur
in solutions of $4x'' + 400x = \cos(\omega t)$? Explain.
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\noindent
IV.  (10) A $4\times 4$ matrix $A$ has eigenvalues $\lambda = 1,2$, each
with multiplicity at least one, and no other eigenvalues.  $A$ satisfies
$\dim\ker (A - I) = 1$ and $\dim\ker (A - 2I) = 2$.
What can you say about the canonical form of $A$?
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\noindent
V.  (15) Find all equilibrium points of the system:
$$\cases{{dx\over dt} = -y(1 - y)&\cr
         {dy\over dt} = -2x(1 + x)&\cr}\leqno(2)$$
\noindent
{\it (over for Extra Credit)}
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\noindent
{\it Extra Credit}
(10) The following is a plot of
a direction field and a solution for some autonomous
first order system.  
\vglue 5.0truein
\noindent
Can this be the system from (2) on previous page?  Explain how you can tell.
\bye
