\documentclass[11pt]{article} \usepackage[psamsfonts]{amsfonts} \usepackage{amsmath,amsthm,amssymb} \usepackage{fullpage} \usepackage{array} \usepackage{supertabular} \renewcommand{\rmdefault}{\sfdefault} \newcounter{lecturectr} \newcommand{\lecture} {\addtocounter{lecturectr}{1} % Day \arabic{lecturectr}} \newcommand{\ds}{\displaystyle} \title{Mathematics 23 Syllabus\\Boyce and DiPrima} \author{The Undergraduate Program Committee} \date{\today} \begin{document} \maketitle \vskip -.2in %\thispagestyle{empty} \newcommand{\PreserveBackslash}[1]{\let\temp=\\#1\let\\=\temp} \let\PBS=\PreserveBackslash \renewcommand{\arraystretch}{2} %\setlength{\extrarowheight}{2pt} \setlength{\tabcolsep}{8pt} \tabletail{\hline} \hskip-.3in \begin{supertabular} % {|>{\PBS\raggedright\hspace{0pt}\parskip=5pt}p{.1\textwidth} % |>{\PBS\raggedright\hspace{0pt}\parskip=5pt}p{.28\textwidth} % |>{\PBS\raggedright\hspace{0pt}\parskip=5pt}p{.55\textwidth}|} {|l |>{\PBS\raggedright\hspace{0pt}}p{.28\textwidth} |>{\PBS\raggedright\hspace{0pt}\parskip=5pt}p{.6\textwidth}|} \hline \textbf{Lecture} &\textbf{Topics/Sections} &\textbf{Some Standard Examples/Concepts} \\ \hline \lecture &2.5 - 2.6 &Modeling: decay, mixing, cooling, growth. Physical models and the differential equations which result. \\ \hline \lecture & 2.1, 2.2, 2.3, 2.4 &Review derivation of solution to first order linear, and review separable equations. Mixing problems (equal rate in/out is separable; unequal rate is FOL) \\ \hline \lecture & 5.2 &Series solutions to first order linear or second order constant coefficient. Solve $y'' + y = 0$ via series, and isolate fundamental solutions as $S(x)$ and $C(x)$. Observe $\sin x $ and $\cos x $ are also solutions. How are $S(x)$ and $C(x)$ related to $\sin x $ and $\cos x $? Leads to representation of functions by series. \\ \hline \lecture & &Define Taylor polynomials; Define geometric series $\sum x^n$ and show when it converges to the rational function $1/(1-x)$; Discuss the notion of Taylor series, and the notion of an interval of convergence: examples $1/(1-x)$ and $\sin x$. \\ \hline \lecture & &Define $p$-series and show when they convrge via an improper integral; state the comparison test and ratio test. Ratio test can be deduced from the comparison test. Use ration test to define the radius of convergence of a power series. \\ \hline \lecture & &Define the notion of absolute/conditional convergence; Discuss alternating series and the error resulting from using partial sums. Consider as an application, the evaluation of (sine, cosine, $e^{-x}$). How could one build a $\sin x$ function for a calculator via partial sums of the Taylor series for the sine? \\ \hline \lecture &3.1 - 3.3 &Review of second order constant coefficient (real roots), \newline Just use characteristic equation and unmotivated solutions here (proofs in a couple of days) \newline [No Wronskian yet] \\ \hline \lecture &3.4 &Review of complex numbers, complex exponential and second order constant coefficient (complex roots). \\ \hline \lecture & 11.3, 11.4 [Crowell \& Slesnick] &Linear Differential Operators (formal ring properties). Homogeneous and nonhomogeneous solutions\newline Set of homogeneous solutions forms a vector space\newline General solution of the form $y_h + y_p$ \\ \hline \lecture &11.4 [Crowell \& Slesnick] &Solve second-order constant coefficient problem by reducing to a first-order system and using general solution to FOL equations. \\ \hline \lecture &4.1, 3.3 &Linear independence of solutions. General definition. \newline Theorem 4.12 [4.1] (w/o Wronskian), that is dimension of space of homog solutions equals the order of the equation. Detecting linear independence: the Wronskian \newline Higher dimensional case after a review of determinants \\ \hline \lecture &7.2 &Review of matrices, determinants (alternating nature) \\ \hline \lecture &4.1 &Wronskian (higher order): problem 20 page 207 \newline Take an nth order linear DE and write it as a system. Introduce matrix notation for this motivating Chapter 7. Perhaps revisit Theorem 4.12. \\ \hline \lecture &3.6 3.7 &[Nonhomogeneous and nonconstant coefficient equations] Finding a particular solution. Undetermined coefficients and variation of parameters. \\ \hline \lecture &3.8/7.1 &Harmonic Oscillators (simple and not) \\ \hline \lecture &7.3 &Systems of linear equations, linear independence, eigenvalues and eigenvectors. \\ \hline \lecture &7.3 &Systems of linear equations, linear independence, eigenvalues and eigenvectors. \\ \hline \lecture &7.4 &Basic Theory of First order systems \\ \hline \lecture &7.5 &Homogeneous linear systems with constant coefficients \\ \hline \lecture &7.6 &Complex Eigenvalues \\ \hline \lecture &7.7, 10.1 &Repeated eigenvalues, Separation of Variables (heat conduction) \\ \hline \lecture &10.1 &Heat equation \\ \hline \lecture &10.2 &Fourier Series \\ \hline \lecture &10.3, 10.4 &Fourier Convergence Theorem; Even and odd functions \\ \hline \lecture &10.4, 10.5 &Even and odd functions; More general heat equation \\ \hline \lecture &10.6 &The Wave Equation \\ \hline \lecture &10.7 &Laplace's Equation \\ \hline \lecture &Wrap it up & \\ \end{supertabular} \end{document}