Spring 2005:

Wk 1:

Mon Jan 16 (Lect 1): Course mechanics, introduction, computer demonstration, introduction to modeling. (Sect 1.1). What it means for a function to be a solution of a differential equation. Guess and Check method. Individual worksheet, Group worksheet. For Wed, fill out the information survey part 1 and part 2.

Wed Jan 18 (L2): Examples of models (worksheet), calculating parameter values in exponential model, how good is the exponential population model (data table), logistic population model and computer graphing.

Wk 2:

Mon Jan 24 (L3): Estimating derivative from a data table. Slope fields (Sect 1.3). Slope field worksheet pt1, pt2. Long term behaviour, sensitive dependence, constant solutions (worksheet).

Wed Jan 26 (L4): Finished with worksheet on long term behaviour, sensitive dependence, butterfly effect and chaos; also constant solutions, attracting (sink) and repelling (source) equilibrium solutions.

Wk 3:

Mon Jan 30 (L5): Homework session.

Wed Feb 1 (L6): Phase line (picture/portrait) (Sect 1.6), Euler’s method (Sect 1.4) with group worksheet.

Wk 4:

Mon Feb 6 (L7): Separation of Variables (Sect 1.2), worksheet. Fishing game. Article about real world fish stock collapses on reserve in the library.

Wed Feb 8 (L8): Review of modeling via exponential model, population models, start of logistic model with harvesting (Sect. 1.7), Harvesting worksheet, Mixing model worksheet. Readings about environmental issues handed out: “Biologists sort lessons of Fisheries Collapse” and “Canada to shield 5 million forest acres” – available in hard copy from Prof. Donnay.

Wk 5:

Mon Feb 13 (L9): Paper review of harvesting write up (feedback rubric), harvesting rewrite instructions. Handout of answer key to mixing problem (paper copy available). Strategies for solving modeling problems.

Wed Feb 15 (L10): People who had studied the same C value compare their results and write up summary. Creating bifurcation diagram (Sect 1.7) and its implications for fishing limits (group worksheet).

Wk 6: Mon Feb 20: (L11): Review for midterm (worksheet).

Wed Feb 22: (L12) Start existence and uniqueness of solutions of differential equations (Sect 1.5). Learn about the harvesting problem of the Tibetan snow lotus plant and the attempt to determine a Wisustainable harvesting level on National Public Radio.

Wk 7: Mon Feb 27 (L13): Formal Existence Theorem and Uniqueness (notes Sect 1.5), Linearity and application to linear differential equations (notes Sect 1.8)

Wed March 1 (L14): Techniques for finding particular solutions to non-homogeneous linear differential equations (Sect 1.8). Method of Undetermined Coefficients = educated guessing! General first order linear system. Constant coefficients and non-constant coefficients. (notes)

Wk 8: Mon March 13^th (L15): Systems of Ordinary Differential Equations (Sect. 2.1, 2.2). Drawing solution curves using HPGSystemSolver . Graphing a vector field (pdf version).

Wed March 15^th (L16): Predator-Prey model derivation. Very simple, uncoupled system; its solutions and phase space. More complicated model with coupling caused by predator-prey interaction. Finding equilibrium solutions. Saddle equilibrium.

Wk 9: Mon March 20 (L17): Comments about upcoming hw assignment and how to interpret the coefficients in a model.

Curve given by parametric equations (x(t), y(t)); its tangent/velocity vector (x’(t), y’(t)). A curve is a solution of the system of differential equations if its tangent/velocity vector matches up with the vector field given by the system.

A non-linear system can be studied by finding equilibrium points and examining phase portrait in neighborhood of equilibrium point. These phase portraits agree with the phase portrait given by a linear system (the linear approximation to the system, where the approximation is based at the equilibrium point). The non-linear phase portrait is created by patching together the linear approximation pieces. For a while we will focus on linear systems; such systems can be written in vector notation using matrices. Interpret a matrix as giving a linear transformation from R² -> R². Worksheet and graph paper.

Wed March 22 (L18): Go over the matrix transformation worksheet. Learn about eigenvectors and eigenvalues (worksheet) (Sect 3.2) and Linearity (Sect 3.1).

Wk 10: Mon March 27 (L19): Connect eigenvalues and eigenvectors to systems of differential equations. Eigenvectors and eigenvalues combine to give particular solutions – the straight line solutions Y₁ and Y₂. Then using linearity of the system, which is based upon a matrix transformation being linear, we showed that the general solution is a linear combination of the straight line solutions: Y(t) = k₁ Y₁(t) + k₂ Y₂(t). Worksheet.

Wed March 29 (L20): Practice using eigenvalues, eigenvectors to find solutions of systems and to draw phase plane pictures by doing a long, multi-step problem.

Wk 11: Mon April 3 (L21): Complex eigenvalues; real and imaginary parts of solutions; spirals. Sect 3.4. Worksheet.

Wed April 5 (L22): More on complex eigenvalues. Repeated eigenvalues (Sect. 3.5).

Wk 12: Mon April 10 (L23): Sect 5.1: Non-linear systems and linearization. Worksheet.

Wed April 12 (L24): Sect 5.1: Linearization via Jacobian matrix, Improved numerical methods (Ch 7). Lab instructions, Euler method, Improved Euler method, Runge-Kutta method.

Wk 13: Mon April 17^th: Improved Euler, Runge-Kutta.

Wed April 19^th: review for midterm.

Wk 14: Monday April 24^th: Mathematical epidemiology: SIR and SIRS models.

Wed April 25^th: surveys; playing with fun programs: Butterfly effect; Lorentz equations, Chemical Osciallations,TD Animations.

Spring 2003

Wk 1:

Tues Jan 21 (Lect 1): Course mechanics, introduction, computer demonstration, introduction to modeling. (Sect 1.1). What is means for a function to be a solution of a differential equation. Guess and Check method.

Thur Jan 23 (L2): Read Sect. 1.1. Guess and Check method (worksheet#0); making a model (i.e. a differential equation) to represent a problem (worksheet#1), terms: 1^st order, 2^nd order, linear, non-linear, linear combination of solutions are also solutions. Population model via exponential function using data.(worksheet #2 CensusData)

Wk 2

Tues Jan 28 (L3): Sect 1.1. What does population model predict? How good are predictions? Read Section 1.3; slope fields. Detools - HPGSolver. Slope Field Worksheet1, SlopeField Worksheet2

Thur Januray 30 (L4): Sect. 1. 6: Equilibrium points and the classification of such points (sink, source). Sect 1.4: Numerical solutions of differential equations via Euler Method. Worksheet for Euler's Method. Vocabulary: 1^st order, 2^nd order; linear, non-linear, autonomous, non-autonomous.

Wk 3

Tues Feb 4 (L5): Review Euler Method and do worksheet (see above). Sect. 1.1 Logistic Population Model, Sect 1.6 Qualitative Analysis using equilibriums, slope function f(P). Graphing all possible solutions on one picture, equilibrium solution, attracting (sink) or repelling (source) equilibrium.

Thur Feb 6 (L6): Sect. 1.2: Mixing Model. Analytic techniquies. Separable equations, solution via integration. Need to review basic techniques of integration.

Wk 4 Lab 3 handout, Euler Method Program

Tues Feb 11 (L7): Solving mixing problems using separation of variables, finding particular solutions, relation with slope field method. . Existence of solutions (Sect. 1.5). Worksheet for Separation of Variables and answer key to worksheet.

Thur Feb 13 (L8): Existence and Uniqueness (Sect 1.5 ). Uniqueness Theorm implies that solutions curves can not cross.

Wk 5:

Tues Feb 18 (L9): Snow, no class.

Thur Feb 29 (L10): Linear DEs. Homogeneous and Non-Homogeneous equations. General solution of the form y = y_h + y_p . Method of Undetermined Coefficients to determine y_p.

Wk 6:

Tues Feb 25 (L11): More on Method of Undetermined Coefficients. Integrating Factor for linear, first order DEs (Sect. 1.8)

Thur Feb 27 (L12): Mixing Problem with variable volume (Sect 1.8), Proof that general solution of inhomogeneous is of the form y = y_h + y_p. Notions of linearity.

Wk 7:

Tues March 4 (L13): Fishing Industry Simulation.

Thur March 6 (L14): (Sect 2.1, 2.2). Bifurcation diagram for logistic map with harvesting (S. 1.7), Gp worksheet for bifurcation diagram.

 Spring Break: Tues March 11, Thur March 13.

Wk 8

Tues March 18 (L15): Introduction to Systems of Equations. Geometric, Numerical approaches. Equilibrium points, meaning and calculations. Predator-Prey models. Interaction term. Physical interpretation of sign of interaction term (Competition, Cooperation). Parametric equations, tangent vectors, system of differential equations, vector fields. Gp worksheet.

Thur March 20 (L16): Review of Bifurcation points (S. 1.8) - see the program PhaseLines. Geometric meaning of a solution of a system of differential equations: rhs = tangent vector to curve; lhs = vector field. These should match. 

Wk 9: Lab 4: Linear Systems

Tues March 25 (L17): (S. 2.3) Analytic solutions of a system; Guess and Check. Linear combinations of a solution is still a solution (S. 3.1). General solution. gps/02gplinearalgebra.pdf

Thur March 27 (L18): (S. 2.1, 2.2, 2.3) Converting 2^nd order DE into first order system: worksheet. Example of 2^nd order equation: Simple Harmonic motion: springs, with friction. Graphing curves x(t) and y(t) separately versus (x(t), y(t)). Program DESketchPad. Writing a system using matrix notation. (S. 3.1)

Wk 10: file://localhost/labs/03lab5%20NonLinear.pdf.

Tues April 1 (L19): Properties of matrices. Superposition principle of solutions, general solution of system. (S. 3.1). Eigenvectors (program MatrixFields). Straight line solutions. (S.3.2). Worksheet to draw solution curve and see straight line solutions.

Thur April 3 (L20): Calculating Eigenvalues, eigenvectors (3.2) Wksheet for eigenvalues, eigenvectors. Answer Key. Relation with straight line solutions, phase portrait (3.3)

Wk 11:

Tues April 8 (L21) More practice with solutions of systems, graphing. Worksheet for solving system using eigenvectors, eigenvalues. Answer Key .

Thur April 10 (L22): Review for mid-term.

Wk: 12

Tues April 15 (L23): Real eigenvalues, same sign, not equal; same sign equal. (S 3.3). Complex numbers review (Appendix B). Complex worksheet. Answer Key complex worksheet.

Thur April 17 (L24): Solutions for complex eigenvalues (S. 4)

Wk 13: Tues April 22 (L25): Rotation matrix as explanation of spiral behaviour in case of complex eigenvalues. Improved Euler Method (7.2). file://localhost/gps/03ImprovedEuler%20.pdf on Improved Euler Method. Computer programs for Euler, Improved Euler and Runge Kutta.

Thur April 23 (L26): Non-linear systems (S 5.1), Linear approximations near equilibrium points, Jacobian Matrix, local phase portrait near equilibrium pt, global phase portrait. Worksheet, answer key to worksheet for Competing Species system.

Wk 14: Lab instructions, Euler method, Improved Euler method, Runge-Kutta method

Tues April 29 (L27): ). Linear approximation near an equilibrium point Local and global phase portraits.

Thur April 30 (L28): Finish theory of linear approximations. Discussion of rates of convergence for Euler, Improved Euler, and Runge Kutta methods of numerical integration (Ch 7). file://localhost/computer%20pgms/NumericalMethodsData.PDF from labs. Quick way to solve 2^nd order equations (3. 6)

"On time and under budget"

Review of partial derivatives (worksheet). Linear approximation near an equilibrium point (old worksheet).

Another old (worksheet).