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\begin{center} {\large April 22, 1998} \end{center}
\begin{center} {\large Ordinary Differential Equations, MATH 325, Exam 3}\end{center}

\vspace{1cm}
{ \hspace{8cm} {\bf Name:}}

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\noindent
This test consists of 5 partial credit problems. It will
be exactly 50 min 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 that your test consists of 6 PAGES with 5 PROBLEMS.   The total
point value of the test is 100 points.
Use the back of the test pages for scratch work.\\
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\noindent
{\bf I have neither given nor received unauthorized aid on this exam:}
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\noindent
1.  (15 points) Suppose that the given vectors $X_1$ and $X_2$ are solutions of a system of linear differential
equations: ${\displaystyle \frac {d X}{dt} = A X}$. Determine whether the vectors form a fundamental set on $(-\infty, \infty)$ 
\[
X_1 = 
\left (
\begin{array}{c}
{1}\\
{\displaystyle -1 }
\end{array}
\right) {\displaystyle e^t}, \;\;\;
X_1 = 
\left (
\begin{array}{c}
{2}\\
{\displaystyle 6 }
\end{array}
\right) {\displaystyle e^t} +
\left (
\begin{array}{c}
{8}\\
{\displaystyle -8 }
\end{array}
\right) {\displaystyle t e^t}.
\]
\newpage

\noindent
(2) (20 points) Find the general solution of the following system: 
\[
X' = 
\left (
\begin{array}{c}
{\displaystyle 3 \;\; -18}\\
{\displaystyle 2\;\; -9} }
\end{array}
\right ) X.
\]

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\noindent
(3) (20 points)
The following system has a single critical point at $(x_0,y_0)$:
 \begin{eqnarray*}
  x' &=& x - 2y  \\
  y' &=& 3x -4y -2. 
  \end{eqnarray*}
Classify it as to type and stability.\\
{\bf Hint}: Begin by finding $(x_0,y_0)$ and by making substitution of the form:\\
$u = x - x_0, v = y - y_0$. 
\newpage

\noindent
(4) (20 points)   Investigate the type and stability of the critical point (0,0) of  the following 
system.  Begin by showing that it is almost linear.
 \begin{eqnarray*}
  x' &=& x + 2y + x^2 + y^2 \\
  y' &=& 2x -2y - 3xy. 
  \end{eqnarray*}
\newpage

\noindent
(5) (25 points) Find all critical points of the following almost linear system and investigate 
the type and stability
of each point: 
  \begin{eqnarray*}
  x' &=& x - y  \\
  y' &=& x^2 -y. 
  \end{eqnarray*}
{\bf Hint}: Assume that the system is almost linear. Do NOT show that it is almost linear.

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