\documentclass[11pt,reqno]{amsart}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Math packages
\usepackage{amscd} %for diagrams
\usepackage{amsfonts} %extra math fonts
\usepackage{amsmath} %miscellaneous enhancements for math formulas
%\usepackage{amsrefs} %pack­age for bib­li­ogra­phies
%for alpha style while using amsart/amsrefs, use the option 
%\usepackage[shortalphabetic]{amsrefs} or [alphabetic], see
%https://tex.stackexchange.com/questions/381082/bibtex-style-alpha-is-not-working
\usepackage{amssymb} %extra math symbols
\usepackage{amsthm} %theorems set-up
\usepackage{latexsym} %extra math symbols
\usepackage{mathrsfs} %\math­scr sommand
\usepackage{mathtools} %pro­vides a se­ries of pack­ages de­signed to en­hance 
%the ap­pear­ance of doc­u­ments con­tain­ing a lot of math­e­mat­ics
%\mathtoolsset{showonlyrefs} %This numbers only equations that are labeled and referred to
       %Use \mathtoolsset{showonlyrefs=true,showmanualtags=true} to show also manual tags
       %For issues with subequations, use \noeqref to silently reference the equations, see:
       %https://tex.stackexchange.com/questions/62037/mathtools-showonlyrefs-fails-with-subequations
\usepackage{slashed}  %Use­ful for the Physi­cist’s ‘Feyn­man slashed char­ac­ter’ no­ta­tion.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Basic general latex packages
\usepackage{verbatim}
\usepackage{cases}
\usepackage[dvips]{epsfig}
\usepackage{epsf} %The orig­i­nal (and now ob­so­les­cent) graph­ics in­clu­sion 
%macros for use with dvips, still widely used by Plain TEX users (in par­tic­u­lar).
%\usepackage{hyperref}
\usepackage[bookmarksnumbered,pdfpagelabels=true,plainpages=false,colorlinks=true,
            linkcolor=black,citecolor=black,urlcolor=black]{hyperref}
\usepackage{url}
\usepackage{color}
\usepackage{xcolor}
\usepackage{graphicx}
\usepackage{enumitem} %for breaks in enumeration:
%http://tex.stackexchange.com/questions/135726/intertext-like-command-in-enumerate-environment
\usepackage{cite} %automatically order references in multiple cites

%More general latex packages
\usepackage{pifont} %com­mands for Pi fonts
\usepackage{upgreek} %up­right Greek let­ters
\usepackage{fancyhdr} %ex­ten­sive fa­cil­i­ties, both for con­struct­ing head­ers and foot­ers, 
%and for con­trol­ling their use
\usepackage{calligra} %cal­li­graphic font in the hand­writ­ing style
\usepackage{marvosym} %Euro cur­rency sym­bol as de­fined by the Euro­pean com­mis­sion, 
%along with sym­bols for struc­tural en­gi­neer­ing; sym­bols for steel cross-sec­tions; 
%as­tron­omy signs (sun, moon, plan­ets); the 12 signs of the zo­diac; scis­sor sym­bols; 
%CE sign and oth­ers.
\usepackage[percent]{overpic} %a cross be­tween the LATEX pic­ture en­vi­ron­ment and 
%the \in­clude­graph­ics com­mand
\usepackage{pict2e} %ex­tends the ex­ist­ing LATEX pic­ture en­vi­ron­ment
\usepackage[hypcap=false]{caption} %option false to avoid warning when hyperref is loaded
\usepackage{wasysym} %The WASY2 (Waldi Sym­bol) font by Roland Waldi pro­vides many 
%glyphs like male and fe­male sym­bols and as­tro­nom­i­cal sym­bols, as well as the 
%com­plete lasy font set and other odds and ends
\usepackage{accents}
\usepackage{float} % used to place images in exact locations 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Margins, spacing, etc.

%EXACT 1in MARGINS with geometry package                         
%geometry package:
%ftp://ftp.fu-berlin.de/tex/CTAN/macros/latex/contrib/geometry/geometry.pdf
\usepackage[papersize={8.5in,11in}, margin=1in,footskip=0.4in,vmarginratio={1:1},hmarginratio={1:1}]{geometry} 
%nohead option from:
%see the documentaion and:
%https://tex.stackexchange.com/questions/318696/1-inch-all-around
%Uselful: option showframe prints the frames on the page for actual visualization
%marginratio is important when using amsart, see:
%https://tex.stackexchange.com/questions/133860/problem-with-margins-using-amsart-and-geometry-packages

%EXACT 1in MARGINS by hand       
%\setlength{\textwidth}{6.5in}     
%\setlength{\oddsidemargin}{0in} 
%\setlength{\evensidemargin}{0in}  
%\setlength{\textheight}{8.8in}   
%\setlength{\topmargin}{0in}      
%\setlength{\headheight}{0in}     
%\setlength{\headsep}{0in}        
%\setlength{\footskip}{.2in}      

%These give wider space on the page
%\setlength{\textwidth}{6.75in} \setlength{\textheight}{8.6in}
%\hoffset=-0.9truein \voffset=-0.1truein
%\baselineskip=7.0mm
%\setlength{\baselineskip}{1.09\baselineskip}
%These are all to reduce blank space between paragraphs, sections etc
%http://www.terminally-incoherent.com/blog/2007/09/19/latex-squeezing-the-vertical-white-space/
%see also
%http://tex.stackexchange.com/questions/108684/spacing-before-and-after-section-titles
%\usepackage[compact]{titlesec}
%\setlength{\parskip}{0pt}
%\setlength{\parsep}{0pt}
% \setlength{\headsep}{0pt}
%\setlength{\topskip}{0pt}
% \setlength{\topmargin}{0pt}
%\setlength{\topsep}{5pt}
%\setlength{\partopsep}{0pt}
%\linespread{0.95}
%\titlespacing{\section}{0pt}{3pt}{1pt}
%\titlespacing{\subsection}{0pt}{3pt}{1pt}
%\titlespacing{\subsubsection}{0pt}{*0}{*0}

%Changing margins in parts of the text:
%http://tex.stackexchange.com/questions/68855/changing-top-bottom-left-right-margins-on-the-fly
%\begin{adjustwidth}{2cm}{2cm}
%text here
%\end{adjustwidth}
%\usepackage{changepage}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Counting, table of contents, etc.
\setcounter{secnumdepth}{4} %depth to which section numbering occurs
\setcounter{tocdepth}{4}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Header, running title
%Comment these to eliminate the running title and authors in the header
\pagestyle{fancy}
\headheight 27pt
\rhead[]{\thepage}
\chead[\textsf{MATH 2610}]{\textsf{Vanderbilt University}}
\lhead[\thepage]{}
\rfoot[]{}
\cfoot{}
\lfoot{}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Theorems, definitions, equations, etc.
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{claim}[theorem]{Claim}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{assumption}[theorem]{Assumption}

\theoremstyle{definition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{notation}[theorem]{Notation}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{convention}[theorem]{Convention}

\newenvironment{dem}{{\noindent \bf Proof: }}{\hfill $\qed$\\ \medskip}
\newenvironment{sketch}{{\noindent \bf Proof sketch: }}{\hfill $\qed$\\ \medskip}

%\numberwithin{equation}{section}
\allowdisplaybreaks
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%General commands

%%%%%%%%%%%%%%%%
\newcommand{\norm}[1]{\Vert#1\Vert}
\newcommand{\abs}[1]{\vert#1\vert}
\newcommand{\nnorm}[1]{\left\Vert#1\right\Vert}
\newcommand{\aabs}[1]{\left\vert#1\right\vert}
\newcommand{\generalfunction}{\mathrm{f}}
\newcommand{\td}{t} %use for the time variable, and if one wants to change it for x^0, adjust here

\newcommand{\dive}{\mbox{\upshape div}\mkern 1mu}
\newcommand{\curl}{\mbox{\upshape curl}\mkern 1mu}
\newcommand{\tr}{\operatorname{tr}}
\newcommand{\vol}{\operatorname{vol}}
\newcommand{\id}{\operatorname{id}}


%to create space among tensor indices
\newcommand{\mss}{\hspace{0.2cm}}
\newcommand{\ms}{\hspace{0.25cm}}
\newcommand{\msm}{\hspace{0.3cm}}
\newcommand{\msb}{\hspace{0.35cm}}
%alternatively, using tensor package, e.g., $M\indices{^\alpha_\beta}$.
\usepackage{tensor}

%Variables and operators in text form: choose between using 
%\mbox, \operatorname, etc.:
%\newcommand{\uperpvort}{\mbox{\upshape vort}}
%\newcommand{\uperpvort}{\operatorname{vort}}
%See:
%https://tex.stackexchange.com/questions/70632/difference-between-various-methods-for-producing-text-in-math-mode
%https://tex.stackexchange.com/questions/48459/whats-the-difference-between-mathrm-and-operatorname/48463
%for differences


%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Enumerating etc.
%https://tex.stackexchange.com/questions/116101/add-bold-enumerate-items
%https://tex.stackexchange.com/questions/23723/latex-enumerate-bold-item-with-non-bold-text
\newenvironment{myenumeratealpha}
{ 
	\begin{enumerate}[label=\textbf{(\alph*)},ref=(\alph*), itemsep=1pt,parsep=0pt,topsep=2pt,labelsep=2pt,wide]
}
{ \end{enumerate}                  } 

\newenvironment{myenumerateroman}
{ 
	\begin{enumerate}[label=\textbf{(\roman*)},ref=(\roman*), itemsep=1pt,parsep=0pt,topsep=2pt,labelsep=2pt,wide]
}
{ \end{enumerate}                  } 

\newenvironment{myenumerateRoman}
{ 
	\begin{enumerate}[label=\textbf{(\Roman*)},ref=(\Roman*), itemsep=1pt,parsep=0pt,topsep=2pt,labelsep=2pt,wide]
}
{ \end{enumerate}                  } 

\newenvironment{myitemize}
{ 
	\begin{itemize}[itemsep=1pt,parsep=0pt,topsep=2pt,labelsep=2pt,wide=6pt]
}
{ \end{itemize}                  } 
%%%%%%%%%%%%%%%%


\begin{document}
%The [Short title] and [Disconzi] will be overriden by the Header, running title commands
\title[MATH 2610]{MATH 2610, Examples of section 4.7}
\author[Vanderbilt]{Vanderbilt University}

\maketitle


\noindent \textbf{Question 1.} 
Find two linearlly independent solutions of
\begin{gather}
x^2 y^{\prime\prime} - 4xy^\prime + 6y = 0.
\nonumber
\end{gather}


\noindent \textbf{Solutions.}
\vskip 0.2cm
\textbf{1.}This is a Cauchy-Euler equation with $a=1$, $b=-4$ and $c=6$. Its
characteristic equation is then
\begin{gather}
a\lambda^2 + (b-a)\lambda + c = \lambda^2 -5 \lambda + 6 = 0,
\nonumber
\end{gather}
whose solutions are
\begin{gather}
\lambda_1 = 2,\, \text{ and } \,  \lambda_2 = 3.
\nonumber
\end{gather}
Therefore 
\begin{gather}
y_1 = x^2, \text{ and } \, y_2 = x^3
\nonumber
\end{gather}
are two linearly independent solutions of the equation.


\bibliographystyle{plain}
\bibliography{References.bib}

\end{document}
