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\begin{document}
 
\baselineskip.525cm
 
\title[Differential Equations]
{Exam 1\\Math 4233 Section 001, Spring 2005}
 
\author[Weiping Li]{Instructor: Weiping Li}
 
\maketitle

{\bf Print Name and Student $\#$}
\vspace{4ex}
\medskip

{\bf SHOW WORK FOR CREDIT !!! MUST PROVIDE NECESSARY WRITTEN STATEMENTS TO SUPPORT YOUR ANSWERS!!!}
\medskip

\begin{enumerate}
\item (10pts) (a) Let ${\bf x}^{(1)}(t)$ and ${\bf x}^{(2)}(t)$ be solutions
of the homogeneous equation ${\bf x}^{'} = P(t){\bf x}$ with initial condition
${\bf x}^{(1)}(0)=(1,-1)$ and ${\bf x}^{(2)}(0) = (-2, 2)$.
Does ${\bf x}^{(1)}(t)$ and ${\bf x}^{(2)}(t)$ form a fundamental set of 
solutions of the system ${\bf x}^{'} = P(t){\bf x}$ ?
\vspace{1in}

(b) State the order $n$ for the global truncation error by using
forward Euler formula, Heun formula and Runge-Kutta formula respectively.
\vspace{1in}

\item (10pts) 
(a) In what intervals are $\left(\begin{array}{c}t^2\\2t \end{array} \right)$
and $\left(\begin{array}{c}e^t\\e^t \end{array} \right)$ linearly independent ?

(b) 
What conclusion can be drawn about the coefficients in the system of homogeneous
differential equations satisfied by these two vectors ?

\newpage

\item (20pts) Find a general solution for the system
${\bf x}^{'} =\left(\begin{array}{cc} 1&-1\\1&3 \end{array}\right)$,
where the coefficient matrix has eigenvalues $r_1=r_2=2$ with a
single eigenvector $(1, -1)$.
\newpage


\item (20pts) Let $A=\left(\begin{array}{cc}-1&-4\\1&-1\end{array}\right)$.
Find $\exp (At)$.
\newpage

\item (20pts) For the system ${\bf x}^{'}=\left(\begin{array}{cc}
1&\alpha \\1 &1\end{array}\right){\bf x}$,

(a) Determine the eigenvalues in terms of $\alpha$.

(b) Find the critical values of $\alpha$ where the qualitative nature of the 
phase portrait for the system changes.

(c) What type of the critical point ${\bf x}=0$ is in terms of different $\alpha$ ?

(d) State the corresponding stability for the critical point ${\bf x}=0$.

\newpage
\item (20pts) Use the method of variation of parameters to solve the 
inhomogeneous system ${\bf x}^{'} = A{\bf x} + {\bf g}(t)$, where the
matrix $A$ has eigenvalue $+1$ with eigenvector $(1, -1)$ and
eigenvalue $-2$ with eigenvector $(2, 1)$, and ${\bf g}(t) =
\left(\begin{array}{c} e^{-t}\\ -e^{t} \end{array} \right).$

\end{enumerate}
\end{document}