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\begin{document}

\begin{center}
{\Large TEST 2

Math 2250}
\end{center}

\begin{itemize}
\item  Show all your work and make your work neat.  You will be graded on this!
\item  You can use a calculator, but I must be able to follow your work as if you had no calculator.
\item  You will be graded on both your answer and the work done to get that answer.
\end{itemize}


\begin{enumerate}

\item  Solve the initial value problem below:
\[
y''-y=0 \qquad y(0)=6 \qquad y'(0)=-4
\]



\newpage

\item  Consider the linear differential equation below:
%
\begin{equation} \label{ode1}
x^3y^{(3)} - 3x^2y''+6xy'-6y= 12x^4-8
\end{equation}
\begin{enumerate}
\item  Show that the following functions are solutions to associated homogeneous differential equation:
  \[
  y_1 = x \qquad y_2=x^2 \qquad y_3=x^3
  \]
\item  Show that the solutions above are linearly independent.
\item  Assume that a particular solution has the form $y_p = Ax^4+B$.
  Determine a particular solution to the non-homogeneous linear differential equation (\ref{ode1}).

\item  Determine the general solution to the non-homogeneous linear differential equation (\ref{ode1}).
\end{enumerate}

\newpage


\item Consider the matrix below:
\[
A = \left[ 
\begin{array}{rrr}
3 & 2 & 1 \\
0 & -3 & 0 \\
0 & 0 & -3
\end{array}
\right]
\]
\begin{enumerate}
\item  Find all eigenvalues of $A$.
\item  Find a basis for each eigenspace of $A$.
\item  If the matrix $A$ is diagonalizable, find a matrix $P$ so that $P^{-1}AP$ is diagonal (and what is this diagonal matrix?).
  If the matrix $A$ is not diagonalizable, explain why not.
\end{enumerate}

\newpage

\item  Transform the following differential equation into a first order system of differential equations.
  Write this system as a matrix differential equation.
\[
t^5 x^{(3)} - x'' \ln t + x' \sin t + x = \tan^2 t
\]

\end{enumerate}

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