### Fall 2008 Syllabus for Math 251

```COURSE DESCRIPTION: Ordinary and Partial Differential Equations (4:4:0)
First- and second- order equations; numerical methods; special functions;
Laplace transform solutions; higher order equations; Fourier series, partial
differential equations. Students who have passed Math 250 may only take a
one credit section of this course.

PREREQUISITE:Math 141

TEXT: Elementary Differetial Equations and Boundary Value Problems, 9th
Edition, Boyce and DiPrima, Wiley and Sons. ISBN: 978-0-470-38334-6

INTRODUCTION

1.1   Direction fields...........................................1/2
1.2   Solutions of Some DE's.....................................1/2
1.3   Classification of DE's.....................................1

FIRST ORDER DE's

2.2   Separable Equations........................................1
2.1   Linear ODE's...............................................1
2.3   Modeling w/DE's............................................4
2.4   Differences Between Linear and Nonlinear Equations.........1
2.5   Autonomous Equations, Population Dynamics..................1
(cover stability and concavity)
2.6   Exact Equations(omit integrating factors)..................1

SECOND ORDER LINEAR EQNS

3.1   Homogeneous Equations with Constant Coefficients...........1
3.2   Fundamental Solutions of Linear Homogeneous Equations      1
3.3   Linear Independence and the Wronskian......................1
3.4   Complex Roots of the Characteristic Equations..............2
(review complex arithmetic)
3.5   Repeated Roots; Reduction of Order using Abel's Formula....3/2
3.6   Nonhomogeneous Equations; Method of Undetermined Coeffs....3
3.8   Mechanical Vibrations (omit electrical vibs)...............3/2
3.9   Forced Vibrations  (w/o damping)...........................1

HIGHER ORDER LINEAR EQUATIONS

4.2   Homogeneous Equations with Constant Coefficients...........1

THE LAPLACE TRANSFORM

6.1   Definition of the Laplace Transform........................2
6.2   Solution of Initial Value Problems.........................2
6.3   Step Functions.............................................1
6.4   DE' w/Discontinuous Forcing Functions......................2
6.5   Impulse Functions..........................................1

SYSTEMS OF FIRST ORDER LINEAR EQUATIONS

7.1   Intoduction to Systems of Differential Equations...........1
7.5-9 Classification of critical pts; sketching phase portraits..4

NONLINEAR DIFFERENTIAL EQUATIONS AND STABILITY

9.1-2 Phase portraits and stability..............................3/2
9.5   Linearize a nonlinear system at each of its................3/2
critical points. Phase portrait for predator-prey eqn

PARTIAL DIFFERENTIAL EQUATIONS AND FOURIER SERIES

10.1  Two Point Boundary Value Problems..........................2
10.2  Fourier Series.............................................2
10.3  The Fourier Theorem........................................1
10.4  Even and Odd Functions.....................................1
10.5  Separtion of Variables; Heat in a Rod......................2
10.6  Other Heat Conduction Problems.............................2
10.7  The Wave Equation: Vibrations of an Elastic String.........2
10.8  Laplace's Equation.........................................2

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