MATH 2403 Differential Equations

Differential Equations Math 2403 F1,F2 Fall 2009

Time and place: TTh: 12:05 - 13:25. Skiles 202.
Text book: Differential Equations by Brennen and Boyce, Wiley 2007
Instructor:

Dr. Luz V. Vela-Arevalo

Office: Skiles 242. Phone: (404)894-6540

Email: luzvela at math gatech edu
Webpage: http://www.math. .gatech.edu/~luzvela Office Hours: Tue, Thu 3-4pm.

Recitation:

Section F1: MW: 13:05-13:55. Instr. Center 217.	Section F2: MW: 13:05-13:55. Skiles 271.
Teaching Assistant: Angel Almada Monter Office: Skiles 146B. Office Hours: Mon, Wed 3-4pm Email: salmada3 at math gatech edu	Teaching Assistant: Yao Li Office: Skiles 225C. Office Hours: Mon 4-5pm, Wed 3-4pm Email: yli at math gatech edu

Homework: Homework will be assigned but will not be collected. However there will be weekly short quizzes. It will be assumed that you understand all the corresponding homework problems for the tests (quizzes, exams, final), hence you are strongly recommended to do all homework.
Attendance: Regular attendance is required both in lectures and recitation sessions.
Make up tests are discouraged. If you have conflicts, you need to tell me as soon as possible and show proper documentation.
Grading: Your final grade is based on the following scheme: 35% your final exam, 40% two midterm exams, 25% weekly quizzes (on Wednesdays, except first week and weeks of midterm exams, or as announced in class). Letter grades will be based on the accumulated points according to the standard cuttofs: A: 90-100%, B: 80-89%, C:70-79%, D: 60-69%, F:0-59%. At the end of the course I will evaluate the class distribution and decide if a curve is needed, which may result only in lowering the above cutoffs. Please note that you cannot use a book, notes, or calculator (or cell phone, or computer, or any other electronic device) during tests.
Course description.
FAQS

1st midterm exam Sep 16

2nd midterm exam Oct 28

Final exam Dec 10 (Thu) 11:30 - 2:20

Course schedule

Week 1. Aug 17-21 Introduction, direction fields, solutions. Euler's method.
Homework: 1.1:1,5,7. 1.2:1(a),2(c),5. 1.3:1(a),1(b),1(c),1(d). 1.4:1,3,5,7,11.

Week 2. Aug 24-28 First order equations. First order linear equations, integrating factors. Separable equations, homogeneous equations. Modeling: mixing tank.
2.1:1,5,7,9,11,13,19,21. 2.2:3,5,7,13,21,23,31,33,35. 2.3:1,3,5.

Week 3. Aug 31-Sep 4 Modeling (cont.): interest loan, escape velocity. Theorem of existence and uniqueness of solutions of differential equations. Autonomous equations. Population dynamics, logistic equation, qualitative analysis and solution.
2.3:7,9,13,16,18,23,24. 2.4:5,7,9,11,13,15,21,24,25. 2.5:1,3,5,7,9,11,13,15,16,20,26.

Week 4. Sep 8-11 (Sep 7 Institute holiday) More population dynamics: critical threshold and logistic equation with threshold. Exact equations. Integrating factors for exact equations. Types of error in numerical methods. Local truncation error for Euler's method.
2.6:3,5,9,11,19,20,25,27. 2.7:1,7,15.

1st midterm exam Sep 16

Week 5. Sep 14-18 Improved Euler's method. Runge-Kutta Method. Review. After first midterm exam: Systems of two first order linear differential equations. Motivation: two interconnected tanks. Homogeneous autonomous systems. Solutions: eigenvalue method.
2.8:1. 3.2:1,3,5,9,11,13,15,24. 3.3:1,3,5,7,9,11,14,16,17,19,25,26.

Week 6. Sep 21-25 (Sep 25 Progress report due) Linear systems with real distinct eigenvalues, general solution and phase portrait. Fundamental set of solutions. Wronskian. Complex eigenvalues, love affair model.
3.4:1,3,5,8,9,13,15.

Week 7. Sep 28-Oct 2 Systems with repeated eigenvalues, generalized eigenvectors. Application to electrical circuits, review of linear systems.
Introduction to nonlinear systems. Numerical methods for systems.
Second order linear equations, some models, theory.
3.5:1,3,5,7,9,15,16. On electrical circuits: 3.2:21,22,23. 3.3:27,28. 3.4:21,22. 3.5:14.
3.6:1,3,5,7,9,11. 3.7:Read example 1. 4.1:1,2,3,4,5,9,11. 4.2:1,3,5,7,9,13,21,23.

Week 8. Oct 7-Oct 9 (Fall Recess Oct 3-6) Abel's theorem for linear systems and linear equations. Linear homogeneous equations with constant coefficients: two real roots, repeated real roots.
4.3: 1,3,5,7,9,11,13,15 (do plots by hand), 19,21,25.

Week 9. Oct 12-16 (Drop day Oct 16) For second order linear equations: Method of reduction of order to find a second solution knowing one; case of complex roots; Cauchy-Euler equations.
Applications: mechanical and electrical vibrations. Undamped free vibrations.
Damped free vibrations: underdamped, critically damped and overdamped vibrations. Method of undetermined coefficients.
4.3: 37,39,41,43. 4.4: 11,13,15,17,19,21,23,25 (do plots by hand), 27, 35,37. 4.5: 1,3,5,7,9,11,13,17,28,29. 4.6: 1,3,5,7,9,11,15,17,19,21.

Week 10. Oct 19-23 Method of undetermined coefficients where repetition exists. Forced vibrations: undamped periodic forcing, cases of superposition and resonance.
Variation of parameters for second order linear equations and for linear systems of dimension 2.
4.6: 23,25. 4.7: 5,7,9,11,17,21. 4.8: 2.5,7,9,15,16,17,19.

2nd midterm exam Oct 28

Week 11. Oct 26-30 Review. After 2nd Midterm Exam: Higher dimensional systems of linear differential equations. Definitions, coupled mass-spring systems, theorem of existence and uniqueness, linear independence and linear dependence of vector functions, Wronskian. Examples, three mixing tanks problem.
6.1: 2,5,7,9. 6.2: 1,3,5,9. 6.3: 1,5,11.

Week 12. Nov 2-6 Complex eigenvalues. Fundamental matrix, exponential matrix. Non homogeneous linear systems. Defective matrices (repeated eigenvalues).
6.4: 1,3,5,7,15. 6.5: 1,3,6,7,13,15. 6.6: 2,3. 6.7: 1,3,5.

Week 13. Nov 9-13 Non-linear differential equations, autonomous systems. Definitions: equilibrium (fixed) point, stable point, asymptotically stable point. Almost linear systems, linearization around a fixed point. Theorem on how the type of stability of the linearization relates to the type of stability of the nonlinear system around a fixed point. Undampded pendulum, damped pendulum.
7.1: Parts (a), (b) and (c) of 1,3,5,13, 17. 7.2: 1,3, 5, 7, 11, 15, 21, 23, 24.

Week 14. Nov 16-20 Competing species: coexistence equilibrium and elimination of one species. Nullclines. Predator-prey model. Limit cycles. Laplace Transform: definition and some examples.
7.3: 1,3,5,10. 7.4: 1,3,5. 7.5: 1,3,5,9. 7.6: Read section. 5.1: 13, 18, 19.

Week 15. Nov 23-25 (Thanksgiving break Nov 26-27)

Week 16. Nov Nov 30-Dec 4 (Last week of classes)

Final exam Dec 10 (Thu) 11:30 - 2:20

Dr. Luz V. Vela-Arevalo
Office: Skiles 242. Phone: (404)894-6540 Email: luzvela at math gatech edu Webpage: http://www.math. .gatech.edu/~luzvela	Office Hours: Tue, Thu 3-4pm.