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\begin{document}






\title{{\large\bf Math 325, Test 2}}
\comment{ \large \begin{minipage}[c]{6.2in} 1. Please cross \fbox{$\times $} the
correct answers. \\ 2. This test will be exactly 50 minutes in length. When you
are told to begin, but not before, glance through the entire test and put your
name on each page. It is YOUR RESPONSIBILITY to make sure your test consists of 5
PAGES with 10 PROBLEMS. The point value for each problem is 10 points with a total
of 100 points. The blank sheet and the backs of the test pages are for scratch
work.\\ 3. On all problems, show your work, indicating clearly in the space
provided how you arrived at your answer.  The points you receive for problems 6 to
10 will depend on the extent to which the work you show convinces the grader (at
the time of grading) that you did all or part of the problem correctly, whether or
not you write down the correct answer. \end{minipage}}
\answersheetfootline{\large\bf \fbox{\rule[-2mm]{0mm}{8mm}Sign your
name:\hspace{6cm}} \hspace{6cm}} \twocolumn{12} 
\date{{\large Nov. 6, 1991}}
\lastversion=0
 \noprofessor \nosection \vertmultchoice=2.5in
\multiplechoiceskip=0.8cm \let\ansborder=\boxborder \boxwidth=1cm \boxdepth=0.4cm
\pagenumberstyle=1
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\problem

 Which picture best describes the trajectories of $x' = Ax$, $ A = \left( \hspace{-2mm}
\begin{array}{cc} 2 & 0\\ 0 & -2\\  \end{array} \hspace{-2mm} \right)$

\vspace{10mm}

a) \hspace{3in}b)

\vspace{20mm}

c) \hspace{3in}d)

\vspace{20mm}

\hspace{3.25in}e)

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\problem 
 Write down a fundamental matrix for $\Phi(t)$ $x' = Ax$,
$A =\left( \hspace{-2mm} \begin{array}{cc} 0 & 1 \\ 1 & 0 \\ \end{array} \hspace{-2mm} \right),$ satisfying
$\Phi (0) = I.$ 




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 \problem 

 a) Show that the vectors $x^{(1)}= \left( \hspace{-2mm}
\begin{array}{c} te^{t} \\ t \\ \end{array} \hspace{-2mm} \right)$ and $x^{(2)} = \left( \hspace{-2mm}
\begin{array}{c}e^{t}\\ 2\\ \end{array} \hspace{-2mm} \right)$ form a linearly independent set
on the interval $[-1, 1]$.

 b) If these vectors satisfy a system $x'=Px$ , what can be said
about the elements of the matrix P?
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 \problem 

 a) Solve the system $x'=Ax$ where $A=\left( \hspace{-2mm}
\begin{array}{cc} 3 & 1 \\ 0 & 3\\ \end{array} \hspace{-2mm} \right)$.

b) Sketch the trajectories of the system. 

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 \problem  
 Find two linearly independent real solutions of
$x'=Ax$ where $A=\left( \hspace{-2mm} \begin{array}{cc} -2 & -1 \\ 7 & 3 \\ \end{array}
\hspace{-2mm} \right)$. 

b) Sketch the trajectories.

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 \problem 
 Consider the system $x'=Ax$ where $A$ has eigenvalues -3
and 1 with corresponding eigenvectors $\left( \hspace{-2mm} \begin{array}{c}1 \\ -1
\\ \end{array} \hspace{-2mm} \right)$ 
and $\left( \hspace{-2mm} \begin{array}{c}1 \\ 2 \\ \end{array} \hspace{-2mm} \right).$ 

a) Sketch the trajectories of this system. 


b) What is the type and stability of the equilibrium point $(0,0)?$



c) Find the fundamental matrix $\Phi (t)$ for the system such that
$\Phi (0) = \left( \hspace{-2mm} \begin{array}{cc}1 & 0 \\ 0 & 1 \\ \end{array} \hspace{-2mm} \right).$

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 \problem 
 The general solution of the homogeneous system $x' =
\left( \hspace{-2mm} \begin{array}{cc}4 & -3\\ 8 & -6 \\ \end{array} \hspace{-2mm} \right) x$ can be shown to
be \\
$x=c_{1}\left( \hspace{-2mm} \begin{array}{c} 3 \\ 4\\ \end{array} \hspace{-2mm} \right)
+ c_{2}\left( \hspace{-2mm} \begin{array}{c}1 \\ 2 \\ \end{array} \hspace{-2mm} \right) e^{-2t}.$ 

a) Using this information and the method of variation of parameters for the
non-homogeneous system $x' = \left( \hspace{-2mm} \begin{array}{cc}4 & 3\\ 8 & 0 \\ \end{array}
\hspace{-2mm} \right) x + \left( \hspace{-2mm} \begin{array}{c} e^t\\ 2e^t \\ \end{array} \hspace{-2mm} \right)$, find the
equations satisfied by the components $u_{1}'$ and $u_{2}'$ of the vector $u'$. 



 b) Find a particular solution of this non-homogeneous system. 
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 \problem

 Find the eigenvalues and eigenvectors for the matrix $ \left( \hspace{-2mm}
\begin{array}{ccc}2 & 0 & 0\\ 0 & 1 & 0\\ 1 & 0 & 3 \\ \end{array} \hspace{-2mm} \right)$
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 \problem 
 Solve the initial value problem $x' = \left( \hspace{-2mm}
\begin{array}{cc} 1 & -4 \\ 4 & -7\\ \end{array} \hspace{-2mm} \right)x, \, x(0) =\left( \hspace{-2mm}
\begin{array}{c} 3 \\ 2\\ \end{array} \hspace{-2mm} \right)$.

\correct $x(t) = \left( \hspace{-2mm} \begin{array}{c} 3 + 4t \\ 2 + 4t \\ \end{array} \hspace{-2mm} \right)
e^{3t}$

\wrong $x(t) = \left( \hspace{-2mm} \begin{array}{c} 3 \\ 2 \\ \end{array} \hspace{-2mm} \right) + \left( \hspace{-2mm}
\begin{array}{c} t \\ t \\ \end{array} \hspace{-2mm} \right) e^{-3t}$

\wrong $x(t) = \left( \hspace{-2mm} \begin{array}{c} 3 \\ 2 t\\ \end{array} \hspace{-2mm} \right) + 
\left( \hspace{-2mm} \begin{array}{c} 1 \\ 1 t\\ \end{array} \hspace{-2mm} \right)e{-3t}$

\wrong $x(t) =  \left( \hspace{-2mm} \begin{array}{c} 3- t \\ 2 + t\\ \end{array}
\hspace{-2mm} \right) e^{3t}$


\wrong $x(t) = \left( \hspace{-2mm} \begin{array}{c} 1 \\ 4 t\\ \end{array} \hspace{-2mm} \right)e^{3t} +
 \left( \hspace{-2mm}
\begin{array}{c} -4 \\ -7 \\ \end{array} \hspace{-2mm} \right)e{-3t}$


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 \problem 
 Which of the following numbers is an
eigen value of 

$A = \left( \hspace{-2mm} \begin{array}{cccc}7 & 2 & 2 & 2\\ 2 & 7 & 2 & 2\\ 2 & 2 & 7 & 2 \\ 2
&2 & 2 & 7 \\ \end{array} \hspace{-2mm} \right)$

Hint: No calculation needed.

\correct $5$

\wrong $1+i$

\wrong $1-i$

\wrong $2$

\wrong $72$

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 \problem  
Consider the system $x'(t) = \left( \hspace{-2mm}
\begin{array}{cc}2 & -5\\ 1 & -2 \\ \end{array} \hspace{-2mm} \right)x.$ Find the fundamental
matrix $\Phi (t)$ satisfying $\Phi(0) = \left( \hspace{-2mm} \begin{array}{cc} 1 & 0 \\ 0 &
1\\ \end{array} \hspace{-2mm} \right)$

\correct $\Phi(t) = \left( \hspace{-2mm} \begin{array}{cc}\cos t + 2 \sin t & -5 \sin t\\ \sin t &
\cos t  - 2 \sin t \\ \end{array} \hspace{-2mm} \right)$ 

\wrong $\Phi(t) = \left( \hspace{-2mm} \begin{array}{cc}\cos t & - \sin t\\ \sin t & \cos
t\\ \end{array} \hspace{-2mm} \right)$

\wrong $\Phi(t) = \left( \hspace{-2mm} \begin{array}{cc}1 + t & t^2\\ t^2 - 2t & 1\\ \end{array}
\hspace{-2mm} \right)$

\wrong $\Phi(t) = \left( \hspace{-2mm} \begin{array}{cc}e^t & e^t + t-1\\ e^{-t}-1 &
e^{-t}\\ \end{array} \hspace{-2mm} \right)$



\wrong $\Phi(t) = \left( \hspace{-2mm} \begin{array}{cc}1 & t\\ -t^2 & e^t\\ \end{array} \hspace{-2mm} \right)$

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  \problem 
 Consider the system
$x' = \left( \hspace{-2mm} \begin{array}{cc}1 & 1\\ 4 & -2\\ \end{array} \hspace{-2mm} \right) + \left( \hspace{-2mm}
\begin{array}{c}e^{-2t}\\ -2e^t\\ \end{array} \hspace{-2mm} \right).$

Find a particular solution:

\correct$\left( \hspace{-2mm} \begin{array}{c}\smallfrac {1}{2}e^t\\ e^{-2t}\\ \end{array} \hspace{-2mm} \right)$

\wrong $\left( \hspace{-2mm} \begin{array}{c}\cosh t\\ 2 \sinh t\\ \end{array} \hspace{-2mm} \right)$

\wrong $\left( \hspace{-2mm} \begin{array}{c}1\\ -4\\ \end{array} \hspace{-2mm} \right)e^{3t} + \left( \hspace{-2mm}
\begin{array}{c}1\\ 1\\ \end{array} \hspace{-2mm} \right) e^{2t}$

\wrong $\left( \hspace{-2mm} \begin{array}{c}5\\ 5\\ \end{array} \hspace{-2mm} \right) e^{2t}$


\wrong $\left( \hspace{-2mm} \begin{array}{c}\cos t\\ -2 \sin t\\ \end{array} \hspace{-2mm} \right)$

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\problem 
 Calculate the asymptotic behavior of $x' = \left( \hspace{-2mm} \begin{array}{cc}2
& -3\\ -4 & 6\\ \end{array} \hspace{-2mm} \right)x$



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\problem
 Consider the system $x^0 = \left( \hspace{-2mm} \begin{array}{cc}0 & 1\\ 0 & 0\\ \end{array}
\hspace{-2mm} \right)x.$


a) Find the fundamental matrix $\Psi(t)$ satisfying $\Phi(0) = \left( \hspace{-2mm}
\begin{array}{cc}1 & 0\\ 0 & 1\\ \end{array} \hspace{-2mm} \right).$


\vspace{5mm}

b) Show that the system is unstable.
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\problem
 Given that $\left( \hspace{-2mm} \begin{array}{c} 1 \\ 0\\ -1 \\ \end{array} \hspace{-2mm} \right)$ is an
eigenvector of $\left( \hspace{-2mm} \begin{array}{ccc} 3 & 2 & 4 \\ 2 & 0 & 2\\ 4 & 2 & 3
\\ \end{array} \hspace{-2mm} \right)$, find the corresponding eigenvalue. 

\vspace{5mm} 

b) Is this eigenvalue of multiplicity 1, 2 or 3? 

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\end{document}