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\title{Mathematics 23 Syllabus\\Boyce and DiPrima}
\author{The Undergraduate Program Committee}
\date{\today}

\begin{document}
\maketitle
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{|l
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\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
&
\\
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