Math
44
Spring
2011
Prof.
Donnay
Class
Materials and Assignments
Wk, Class#, Date 
Class
Material 
Assignment 
Wk 1: Class 1 M 1/17 
Introduction to Course. Problems Facing the World. 
For Wed Jan 19: Read the course
syllabus. What is the Òspecial eventÓ about?  Fill in Student Information Survey  Take Precourse Self Assessment and Review 
C2 W 1/19 
Intro Differential Equations program.
On Blackboard
Discussion Post, write about one of the programs in the Differential
Equations disk. Mathematical Modeling: Finding a
function that matches data. Using the function to predict the future. Pfaff
data set on atmosphere taken from Tom PfaffÕs website.
Question set for CO_{2} data. Excel
instructions on curve fitting (PC version, Mac version).

For Friday Jan 21: a.
Post writing about diff eqns
program b.
Complete CO_{2}
data set assignment. 
C3 F 1/21 
Review of CO_{2} data set assignment.
Exploration of DETools software: Butterfly effect (Aidan); Chemical
Oscillations (Anna). Differential
Equations and exponential functions worksheet. 
For Wed Jan 26, Homework assignment. Need Pfaff module on Global Temperature. The data on
temperature is contained in the same excel
file that we used for the CO_{2} levels. Note: use the
temperature data that starts in 1950. 
C4 W 1/26 
Solving initial value problem for
basic population model dP/dt
= kP using Ôguess and checkÕ method: P(t) = c e^{kt}. ÒGoodÓ
choice of variables. Strengths and weakness of model (worksheet) as
applied to US population modeling.
The modeling cycle: simple assumption, translate in DE, solve DE using
initial data, predict new values, check to see if predictions are accurate,
revise model (if needed) by taking more factors into account. Translating basic assumption about
model (as given in a word statement) into a D.E.: (worksheet). 
For Friday Jan 28: 
do
second problem from translation worksheet. 
Be prepared to (briefly)
describe your Pfaff module and why it is an interesting topic. Part
1 of HW due Wed Feb 2. 
C5 F 1/28 
Vote on which 3 Pfaff modules will be
available for HW. Winners: CO2 emission; Photovoltaic; Ozone Hall. 
For Monday Jan 31: Finish slope fields
worksheet. and bring to class. Part
2 of HW due Wed Feb 2. 
C 6 M 1/31 
Sect 1.3: Review of slope fields
given by f(t,y). Autonomous DE when slope field is f(y); slopes are constant for fixed y, varying t. Nonautonomous
DE when slope field of form f(t) or f(t,y); i.e. slope field depends on t.
When slope field =f(t), slopes constant for fixed t
as y varies. Decide f
type ={f(y), f(t), f(t,y)} for S1.3 #15(iviii). Meaning of
solution of DE via slope fields. Slope yÕ(t) of
solution curve must match up with slope of slope field for y(t) to be a
solution. (handout). Start of Separation of Variables
technique (Sect 1.2). 
Note: Correction: the problems from Sect 1.1 should be #6, 10, 11,
12ab. The problems listed on Part 1 of HW, # 4, 8, 9, 10ab were
from the wrong edition of the text. If you have already done the original
problems, that is fine. If you have not yet done the problems, please do the
new set of problems. 
C 7 W 2/2 
Separation of variables: dP/dt = (15P)/20 . Solve for
general solution. Find particular solution P(0) =
20, 10, 15 and P(2) = 3. Graph solutions. Exponential decay. Shifting graphs
up/down; reflecting. Model a mixing
problem. What to do when you do not know what to do. 
For
Monday Feb 7^{th}: Find the
general solution of
the DE dy/dt = y^{2}. Find
the particular solution that solves the initial condition y(t=0)
= 1. Sketch this solution. For what value of t does the solution Òblow upÓ? For
Wed Feb 9^{th}: HW Wk 3. 
C 8 M 2/7 
Sect 1.5 Existence and Uniqueness. 
For Wed Feb 9^{th}: Quiz 1 
C 9 W 2/9 
Applications of Uniqueness theorem. Sect 1.4 Euler Method: worksheet. Excel spreadsheet
for Euler Method (version 1,
version 2, instructions). 
For Wed Feb 16^{th}: Hw Wk 4. 
C 10 M 2/14 
Continued with Euler Method using
Excel. Equilibrium solutions, slope fields, slope function f(y), phase line (phase line worksheet). 
For Wed Feb 16^{th}: Quiz 2 
C11 W 2/16 
Qualitative analysis of DEs (Sect 1.6) 
HW
Wk 5. Some of the hw is due (linearity) on Monday; the rest due on Wed. 
C12 M 2/21 
Linear differential equations (Sect
1.8) 
For Wed Feb 23^{rd}: Quiz 3 
C13 W 2/23 
DE Challenge Problems.
Solutions of linear DE. y_{g}=y_{h} + y_{p}_{.
}Finding y_{p} via Method of
Undetermined Coefficients (Educated Guess). Sect 1.8. Finance Problem. Using
phase line analysis to study
fishing problem. 
HW
Wk 6: Most due on Wed but a bit is due on Monday. 
C14 M 2/28 
Sect. 1.7. Bifurcation diagram for
Logistic Map with Harvesting worksheet.

For Wed March 2: Quiz 4 
C15 W 3/2 
Integrating Factor Method (Sect 1.9).
Introduction to Systems. 
HW
Wk 7: This will be due on Monday March 14^{th}. It will be
returned by Wed March 16^{th} so you can have it to study for the
Take Home Exam. 
C16 
Discussion
of midterm exam structure (List of topics, What makes a good
answer). Systems of DEs: vector fields (worksheet1),
parametric equations; geometric interpretation. Equilibrium solutions.
Analytic solutions of system. 
For
Wed 3/16. Post questions and potential exam problems to google docs Midterm Review
Document. Do What makes a good
answer worksheet. 
C 17 W 3/16 
Review of midterm. PredatorPrey
equations (S 2.1) 
Take home Midterm given out. Due at
the start of class on Monday 3/21. 
C 18 M 3/21 
Parametric Equation of line; Graphing
parametric equations vs (t, x(t)) graph. Equilibrium
solutions. Physics problems and associated 2^{nd} order DE.
Connection with system. Sect 2.1, 2.2. Mathematica to find
Eigenvalues, Eigenvectors of matrix. 
For Friday 3/25: Worksheet on parametric equations. Textbook
problems. For Monday 3/28Worksheet: Relation between type of
equilibrium solution and eigenvalues. 
C 19 W 3/23 
Harmonic (S. 2.2) and Damped Harmonic
Oscillator (S. 2.3). Decoupled and Partially Coupled Systems (S. 2.3).
EulerÕs Method for Systems (S. 2.4) 
For Wed March 30, HW Wk 9. 
C20 M 3/28 
Course project information.
Vector spaces (hand
out); Conjectures about relationship between eigenvalues and type of
equilibrium; Linear map from R^{2 } > R^{2} given by matrix A; geometric meaning of
eigenvector, eigenvalues (woksheet, graph
paper). 
Quiz
for Wed March 30. 
C21 W 3/30 
Complex numbers; Euler formula, Polar
form. Matrix is linear, linear combination of
solutions of a linear system is a solution. Straight line
solutions with eigenvectors, eigenvalues. General solution. 
For Monday April 4;
post write up about project to Blackboard discussion board. For Wed April 6, HW Wk 10. 
C22 M 4/4 
Practice drawing phase diagrams with
for linear systems with real eigenvalues. Complex eigenvalues give rise to spiral
motions. Mathematica can solve systems
of equations. 
Quiz
for Wed April 6^{th}. 
C23 W 4/6 
Determining the bending for attractors/repellers. Complex eigenvalues give rise to real
solutions. 
HW
Wk 11 including complex number
calculations. Read Project
Guidelines. 
C24 M 
Calculations of eigenvalues/vectors.
Nonlinear cases on computer. 
Quiz
due Wed April 13^{th}. 
C25 W 4/13 
Nonlinear system analysis: find linear
approximation matrix near equilibrium point (Sect 5.1). 3 ways including Jacobian matrix. 
HW
Wk 12 due Wed April 20^{th}. 
C26 M 4/18 
Putting linear pieces together to
make nonlinear system. Phase portrait for diagonal matrix. 
Quiz
due Wed April 20^{th}. 
C27 W 4/20 
Numerical methods: Euler, Improved
Euler, RungeKutta (Sect 7.17.3). Numerical solutions with Mathematica.
(pdf
version) 
HW Wk 13: due Wed April 27^{th}.
Sect. 7.2 #1  Do both Euler and
Improved Euler for this example. Also solve for the
exact value of the integral. Compare. 





