%Homogeneous ODEs with Constant Coefficients %Math340 \documentclass[12pt]{article} \usepackage{times} \pagestyle{empty} \addtolength{\textwidth}{1.2in} \addtolength{\textheight}{1.2in} \addtolength{\oddsidemargin}{-.58in} \addtolength{\evensidemargin}{-.58in} \renewcommand{\baselinestretch}{1.0} \parindent = 0cm \parskip = .1cm \begin{document} \begin{center} {\Large Homogeneous ODEs with Constant Coefficients \footnote{Copyright \copyright 2002 Steve Chapin and Larry Snyder. All rights reserved. Please address comments to young@math.ohiou.edu.}} \end{center} Try the following in \textsc{MatLab}:\\ \verb& syms m& \\ \verb& eqn1 = 'm^2 - 3*m-1 = 0'& \\ \verb& eqn2 = 'm^4 - 4*m^3 + 14*m^2 - 20*m + 25 = 0'& \\ \verb& solve(eqn1)& \\ \verb& solve(eqn2)& \bigskip For each of the following differential equations: \begin{itemize} \item Write down the auxiliary equation. \item Write down, in standard mathematical notation, all of the solutions to the auxiliary equation. (Use \textsc{MatLab} to find the solutions.) \item Write down the general solution of the differential equation. \end{itemize} \bigskip (a) \qquad $y''' + y'' - 6y' - 18y = 0$ \bigskip (b) \qquad $y^{(4)} - 2y''' - 6y'' + 16y' - 8y = 0$ \bigskip (c) \qquad $y^{(4)} - 3y''' + 7y'' + 21y' - 26y = 0$ \bigskip (d) \qquad $y^{(5)} - 2y^{(4)} + 2y''' - 4y'' + y' - 2y = 0$ \bigskip (e) \qquad $2y^{(5)} - y^{(4)} - 4y''' + 3y'' - 8y' - 12y = 0$ \end{document}