% Simplified version of an UMTYMP template.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
% If the text is too small, change 10pt to 12pt
\documentclass[11pt]{article}
\RequirePackage{amsmath,amsthm,amssymb,latexsym,amsthm,verbatim,xspace,amstext,xspace,setspace}

% Here's where you edit the class, section and title information

% Set a fixed margin along all sides
\usepackage[margin=1in, nohead, pdftex]{geometry}
\usepackage{enumerate,graphicx}
% Re-adjust the top margin if desired
\topmargin -0.2in
% These packages used for spacing in the header and throughout



% Uncomment one of the following to set the spacing to single, 1.5 or double spacing
\singlespacing
%\onehalfspacing
%\doublespacing

% This prevents paragraphs from begin indented.  Comment it out to have indentation.
\parindent 0ex
\setlength{\textheight}{9.5in}
% Finally we start!
\begin{document}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% This is the "header" at the top.  You shouldn't have to edit these lines!
% Instead, chance the Class, Section and Title above.
\begin{center}{ \Large{Worksheets}  \\\large{05/01/2018} }\end{center}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


% At this point, you might have some introductory text, or jump straight
% into an ``enumerate'' environment for the problem statements.
\bigskip
\begin{enumerate}
\item Solve the following initial-value problem using {\bf Laplace transform}
\[{\vec{\mathbf{x}}}'=\left[ \begin{matrix}
   3 & -3  \\
   2 & -2  \\
\end{matrix} \right]\vec{\mathbf{x}}+\left[ \begin{matrix}
   4  \\
   -1  \\
\end{matrix} \right],~~~~\vec{\mathbf{x}}(0)=\left[ \begin{matrix}
   1  \\
   0  \\
\end{matrix} \right]\]
\newpage
\item Solve the following initial-value problem using {\bf undetermined coefficient method} (i.e. superposition method)
\[\vec{\mathbf{x}}'=\left[ \begin{matrix}
   0 & -1  \\
   3 & 4  \\
\end{matrix} \right]\vec{\mathbf{x}}+\left[ \begin{matrix}
   t  \\
   -4t-2  \\
\end{matrix} \right],~~~~\vec{\mathbf{x}}(0)=\left[ \begin{matrix}
   1  \\
   1  \\
\end{matrix} \right]\]
\newpage
\item Determine a closed-form solution for the Fibonacci iterative equation
\[{{y}_{0}}=1,~{{y}_{1}}=1,~{{y}_{n+1}}={{y}_{n}}+{{y}_{n-1}}\]
\vspace{3.5in}
\item A certain horse runs in a three-horse race. If he wins a race, he has probability 1/2 of winning, 1/3 coming in second, and 1/6 coming in third the next race. If he comes in second on a race, he has probability 1/3 of winning, 1/6 coming in second, and 1/2 coming in third the next race. If he comes in third on a race, he has probability 1/3 of winning, 1/3 coming in second, and 1/3 coming in third the next race. Let $x_n$, $y_n$, $z_n$ respectively be the probability the horse coming in first, second, third in the n'th race. Then
\[\left[ \begin{matrix}
   {{x}_{n+1}}  \\
   {{y}_{n+1}}  \\
   {{z}_{n+1}}  \\
\end{matrix} \right]=A\left[ \begin{matrix}
   {{x}_{n}}  \\
   {{y}_{n}}  \\
   {{z}_{n}}  \\
\end{matrix} \right]\]
\begin{enumerate}
\item Find the 3$\times$3 matrix $A$.
\newpage
\item The horse comes in second on the first race. Find the probability that he wins / comes in second / third on race $n$. [Given that the eigenvalues of matrix $A$ are 1, 1/6, -1/6.]\vspace{7in}
\item What is the limit of $x_n$ as $n\to\infty$ ?
\end{enumerate}\newpage
\item Solve the initial-value problem
\[\vec{\mathbf{x}}'=\left[ \begin{matrix}
   -1 & 0 & 0  \\
   0 & 0 & 2  \\
   0 & -2 & 0  \\
\end{matrix} \right]\vec{\mathbf{x}},~~~~~~\vec{\mathbf{x}}(0)=\left[ \begin{matrix}
   1  \\
   0  \\
   1  \\
\end{matrix} \right]\]
\end{enumerate}
\end{document}











