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 “special event” about?

-           Fill in Student Information Survey 

-           Take Pre-course 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 Pfaff’s website. Question set for CO2 data. Excel instructions on curve fitting (PC version, Mac version).

 For Friday Jan 21:

a.         Post writing about diff eqns program

b.        Complete CO­2 data set assignment.

C3

F 1/21 

Review of CO2 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 CO2 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 ‘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 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.

Logistic Model derivation. Computational exploration worksheet for logistic equation. Calculating and plotting slope fields worksheet.

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. 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).

 

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 = (15-P)/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 7th: Find the general solution of  the DE dy/dt = y2. Find the particular solution that solves the initial condition y(t=0) = 1. Sketch this solution. For what value of t does the solution “blow up”?

 

For Wed Feb 9th: HW Wk 3.

C 8

M 2/7

Sect 1.5 Existence and Uniqueness.

 

For Wed  Feb 9th: 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 16th: 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 16th: 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 23rd: Quiz 3

C13

W 2/23

DE Challenge Problems. Solutions of linear DE. yg=yh + yp. Finding yp 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 14th. It will be returned by Wed March 16th so you can have it to study for the Take Home Exam.

C16
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 google docs Midterm Review Document. Do What makes a good answer worksheet.

C 17

W 3/16

Review of midterm. Predator-Prey 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 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.

C20

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

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 6th.

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
4/11

Calculations of eigenvalues/vectors. Non-linear cases on computer.

Quiz due Wed April 13th.

C25 W

4/13

Non-linear system analysis: find linear approximation matrix near equilibrium point (Sect 5.1).  3 ways including Jacobian matrix.

HW Wk 12 due Wed April 20th.

C26 M

4/18

Putting linear pieces together to make non-linear system. Phase portrait for diagonal matrix.

Quiz due Wed April 20th.

C27 W

4/20

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.