Math 210

Spring 2011

Prof. Donnay


Class Materials and Assignments


Wk, Class#,


Class Material



Wk 1: Class 1

W 1/19


Introduction to Course. Problems Facing the World.   


Intro Differential Equations program.


For Friday Jan 21 at noon:

-           Fill in Student Information Survey 

Take Pre-course Self Assessment and Review

For next class: Read the course syllabus. What is the “special event” about?


For Monday 1/24. Write about one of the programs on the CD. Post your write up (1 to 2 paragraphs in the Discussion Board of Blackboard).


NOTE: EXTENSION OF THIS ASSIGNMENT TILL WEDNESDAY. On Monday from 6 – 8pm, the math computer lab Rm. 349 will be open. The DETools software is on the computers there.



M 2/24



Pfaff data set on atmosphere taken from Tom Pfaff’s website. Question set for CO2 data. Excel instructions on curve fitting (PC version, Mac version).


Differential Equations and exponential functions worksheet.


 Wed Jan 26th.

a.         Write up neatly (complete sentences, clearly define what your variables stand for and what your units are) and hand in the CO2 assignment.

b.        Post your write up about the DETools module to Discussion Board in Blackboard by noon Wed. 

c.         Hand in the completed Differential Equations and exponential functions worksheet.



W 2/26


Examine DETools Modules: Predator – Prey, Chemical Oscillation (terms: Initial value, circular/periodic behavior, parameters, tipping point = bifurcation value).


Solving initial value problem for basic population model dP/dt = kP using ‘guess and check’ method: P(t) = c ekt. “Good” 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 Monday Jan 28th: Do (as much as you can) the worksheet on translation; we will go over this in your groups on Monday and work further on it.


For Wed Feb 2: Homeworks Assignment. 

You will need the Pfaff module on Global Temperature. The data on temperature is contained in the same excel file that we used for the CO2 levels. Note: use the temperature data that starts in 1950.


Correction: the problems from Sect 1.1 should be #6, 10, 11, 12ab. The problems listed on the HW sheet, # 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.


M 1/31

Modeling: translate statement into DE (worksheet).  Logistic Model derivation. Computational exploration worksheet for logistic equation. Calculating and plotting slope fields worksheet.



W 2/2

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. Non-autonomous 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(i-viii). Meaning of solution of DE via slope fields. Slope y’(t) of solution curve must match up with slope of slope field for y(t) to be a solution. (handout). 


Separation of Variables technique.

For Monday Feb 7th: Do the first problem from the HW set.


For Wed Feb 9th: Do HW Wk 3.


Pfaff Modules: Do one of:

World Oil Consumption       

Ozone Hole                                                                  

World Population 


The data table is on the Pfaff web site. Next to the data table is a word document that gives you the instructions of what you should do with the data. 


M  2/7

Separation of variables worksheet; plotting exponentials; what to do when you do not know what to do worksheet to solve modeling problem about mixing.

For Wed Feb 9th: do quiz 1


W 2/9

Sect 1.5 Existence and Uniqueness.


For Mon Feb 14 and Wed Feb 16:

HW Wk 4


M 2/14

Applications of Uniqueness theorem.

Sect 1.4 Euler Method: worksheet. Excel spreadsheet for Euler Method (version 1, version 2, instructions).

For Wed Feb 16th; Quiz 2


W 2/16

Partial derivative calculations. Qualitative analysis of DE worksheet, phase line. Sect 1.6

HW Wk 5

Note that there are two problems due on Mon Feb 21; the rest are due on Wed Feb 23.


M 2/21

Phase line: Linearization Theorem S 1.6. Introduction to S. 1.8: y­g = yh + yp .


For Wed Feb 23rd: Quiz 3

Guidelines for redoing Quiz 2


W 2/23

Using phase line analysis to study fishing problem.

Sect 1.8: Theorems: Linear combination of solutions of homogeneous equation are solutions. Difference of particular solutions to nonhomogeneous equation is a solution to homogeneous equation.

HW Wk 6: Most due on Wed but a bit is due on Monday.

Here are the DE Challenge Problems. Do two out of the three.


M 2/28

Feedback on Quiz 3.

Sect 1.8: How to find a particular solution to linear nonhomongeneous DE: Educated guess (Method of Undetermined Coefficient for constant coefficient DE).

Sect 1.7: Bifurcation diagram for Logistic Map with Harvesting. Calculate phase line for various c values.

For Wed March 2: Quiz 4



W 3/2

Sect. 1.7. Calculating equilibrium solutions using general formula. Visual method of determing bifurcations. Finance problems. Intro to DE systems using DE Tools.

HW Wk 7: This will be due on Monday March 14th. It will be returned by Wed March 16th so you can have it to study for the Take Home Exam.

Quick write (to be included in the hw). What was one thing that your found interesting about our work on the Bifurcation diagram? What is one question you have about bifurcations?  


M 3/14

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 Blackboard Discussion board. Do What makes a good answer worksheet.


W 3/16

Review for midterm.

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 2nd 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). Euler’s Method for Systems (S. 2.4)

For Wed March 30, HW Wk 9.


M 3/28

Course project information. Vector spaces (hand out); Conjectures about relationship between eigenvalues and type of equilibrium; Linear map from R2  -> R2 given by matrix A; geometric meaning of eigenvector, eigenvalues (woksheet, graph paper).

Quiz for Wed March 30.

C21 W 3/30

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


Complex numbers. Polar form. Euler’s formula. 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 6th.

C23 W


Determining the bending for attractors/repellors. Complex eigenvalues give rise to real solutions.

HW Wk 11 including complex number calculations. Read Project Guidelines.



Instructions for re-assessement of topics from midterm. To help organize the reassessment process, you should fill in this reassessment sheet and hand in to Professor Donnay when you hand in your preparatory work.

C24 M

Calculations of eigenvalues/vectors. Non-linear cases.

Quiz due Wed April 13th.

C25 W


Non-linear system analysis: find linear approximation matrix near equilibrium point (Sect 5.1).

HW Wk 12 due Wed April 20th.

C26 M


Jacobian matrix of partial derivatives. Putting linear pieces together to make non-linear system. Phase portrait for diagonal matrix.

Quiz due Wed April 20th.

C27 W


Numerical methods: Euler, Improved Euler, Runge-Kutta (Sect 7.1-7.3). Numerical solutions with Mathematica. (pdf version)

HW Wk 13: due Wed April 27th.

Sect. 7.2 #1 - Do both Euler and Improved Euler for this example. Also solve for the exact value of the integral. Compare.