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, Eighth
Edition, Boyce and DiPrima, Wiley and Sons. ISBN: 0 471-33338-1
SECTION TOPICS PERIODS
INTRODUCTION
1.1 Direction fields 1
1.2 Solutions of Some DE's 1/2
1.3 Classification of DE's 1/2
FIRST ORDER DE's
2.1 Linear Equations with Variable Coefficients 1
2.2 Separable Equations 2
2.3 Modeling with First Order Equations 3
(do mixture, interest and air resistance)
2.4 Differences Between Linear and Nonlinear Equations 1
2.5 Autonomous Equations and Population Dynamics 1
2.6 Exact Equations (omit integrating factors) 1
SECOND ORDER LINEAR EQNS
3.1 Homogeneous Equations with Constant Coefficients 2
(cover the equations with missing y or missing t,
show how to solve initial value problems with data
is specified not at 0)
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
(also review complex arithmetic) 2
3.5 Repeated Roots; Reduction of Order 1
3.6 Nonhomogeneous Equations;
Method of Undetermined Coefficients 3
3.8 Mechanical Vibrations (omit electrical vibs) 2
3.9 Forced Vibrations (no damping) 1
THE LAPLACE TRANSFORM
6.1 Definition of the Laplace Transform 2
6.2 Solution of Initial Value Problems 2
6.3 Step Functions 2
6.4 Differential Equations 1
with Discontinuous Forcing Functions
6.5 Impulse Functions 1
SYSTEMS OF FIRST ORDER LINEAR EQUATIONS
(This chapter and Chapter 9 must be
filtered extensively. See note below!)
7.1 Intoduction to Systems of Differential Equations 1
7.5-9 Classification of critical points and sketching 3
phase portraits.
NONLINEAR DIFFERENTIAL EQUATIONS AND STABILITY
9.1 Phase portraits and stability 1
9.2 Phase portraits for Nonhomogeneous Linear systems 1
9.5 Linearize a nonlinear system at each of its
critical points. Phase portrait for predator-prey eqn 1
PARTIAL DIFFERENTIAL EQUATIONS AND FOURIER SERIES
10.1 Two Point Boundary Value Problems 1
10.2 Fourier Series 2
10.3 The Fourier Theorem 2
10.4 Even and Odd Functions 2
10.5 Separtion of Variables; Heat in a Rod 2
10.5 Other Heat Conduction Problems 3
10.6 The Wave Equation: Vibrations of an Elastic String 1
10.7 Laplace's Equation 1
Review Periods 3
----
Total number of periods 56
NOTES:
1. While direction fields can be covered using just the examples in the
text, easy to use direction field plotting software is available for
the TI graphing calculators. Furthermore, a superb piece of software
called dfield, which draws direction fields and trajectories (with
initial condition defined by a click of the mouse), is freely
available. Dfield runs under Matlab but requires absolutely no
knowledge of Matlab to use. Also see the phase portrait
Java applet at: http://www.math.psu.edu/melvin/phase/newphase.html
2. Java applets and TI software to be used for the numerical solutions
sections available at: www.math.psu.edu/glasner/m251/ Limit the comparison
of order of accuracy of these methods to demonstrating them experimentally
in a specific example.
3. Boyce and DiPrima treat n x n linear systems and thus develop a fair
amount of linear algebra, most of which is not needed or trivial for 2 x 2
systems. It not easy to extract what is needed for the 2 x 2 case from their
exposition. The treatment of 2 x 2 linear systems x' = ax + by, y' = cx + dy
in Simmons (Differential Equations with Applications and Historical Notes
published by McGraw Hill) formally avoids linear algebra. The idea is to
guess the solutions x = A exp(rt) and y = B exp(rt), and then the problem
reduces to studying the system of algebraic equations (a - r)A + bB = 0, cA
+ (d - r)B = 0. The classification of critical points and sketching of phase
portraits can be done as by Simmons. There are two approaches: evaluate
partial derivatives at critical points, or, move critical points to origin
and knock off nonlinear terms. Sketching the phase portrait for the
predator-prey equation should be the main goal here.