Introduction to (Elementary) Differential Equations
Math 305-3
Spring 2003

Textbook: Elementary Differential Equations and Boundary Value Problems, 7^th Edition. by William E. Boyce and Richard C. DiPrima, John Wiley & Sons, Inc., New York, 2001.

Read this Without Tears: WHAT IS EXPECTED OF YOU, see "Teaching at the University Level" by Stephen Zucker, Notices Amer. Math. Soc. 43 (1996), p. 863.

Course Description: This course is a very elementary introduction to the solution techniques, the theory and applications of differential equations, emphasizing second order ordinary differential equations (ODEs), Heat and Wave equations (as linear PDEs) and illustrated by many examples. It has the form of a more calculus-based course on computing solutions, instead of a more theoretically profound course on this subject.

Prerequisites and Development of Contents: This course should be accessible to any student with a $C-$ in Calculus II (i.e. Math 250 and, hopefully, also in Applied Linear Algebra). The content of this course itself should very nearly coincide with that of our textbook, excluding most of the parts on numerical methods (unfortunately), excluding chapters 6 - 9, 11. Thus, we will treat first order equations, 2nd order and higher order equations, series solutions, Fourier series and simple PDEs like Heat and Wave equations. I am not perfect. However, be sure that I will do my very best to please you and your expectations.

Readings, Problem Sets, Exams: Readings and problem sets will be from the text and my manuscript, and it will be assigned in classes and perhaps additionally published at my homepage. Exams will cover all material covered in the lectures and/or the readings. The authors seem to have put a fair effort into his presentation from very practically oriented point of view, and his approach is probably quite different from what you've seen before. Thus, I really encourage you to read the book and, hopefully, additional related literature, as lectures advances.

Exam Dates:

• Midterm I: Friday, February 21, 2003 (in class)
• Midterm II: Friday, March 21, 2003 (in class)
• Midterm III: Friday, April 25, 2003 (in class)
• Final Exam: Friday, May 9, 2003, 12:50 pm - 2:50 pm (AG 170)

Grading Strategies - Grade Distribution:

• 25 % Homeworks & Quizzes, 50 % Midterms, 25 % Final Exam
• 90 % <= A-class, 75 % <= B-class, 60 % <= C-class, 45 % <= D-class
• You may choose either a Computational Project (Programming of Numerical Methods & Dynamical Systems) or a Theoretical Project (Qualitative Behavior of Dynamical Systems, Systems of ODEs or PDEs) to substitute the worsest parts or any desired sections of your grade distribution (but not more than 25% of total grade). Projects need to be started before April 15, 2002, and handed in to the instructor in a written form by April 30, 2002. We will keep your project report for data purposes, so do a copy before. An oral presentation concerning the project results must follow by May 3, 2002.
• Your scores are recorded in my office, and evaluated according to the gradeline distribution given. Someone (graduate student at Math Department) is your grader for homeworks. I will grade the exams.
• A total of 16 homeworks is due at the end of the semester. We will take the best 15 homeworks individually for your overall grade.

Course Syllabus (Last Update: 01/10/02) - You will need an utility like ghostview to read it!

Please, note you need ghostview to read the postscript files after downloading! See http://www.cs.wisc.edu/~ghost/index.html for more software product information.

Remarks for Withdrawal:

The last day to withdraw from the course without a grade is Friday, January 24. The last day to withdraw from the course with a grade of W is Monday, March 17. You should be aware that the grade INC (incomplete) cannot be used as a withdrawal and can only be assigned to students who are passing the course and for reasons beyond their control cannot complete all class assignments. See the undergraduate catalog for more details.

Remarks for the Prerequisites and First Week:

Please, review your knowledge on Calculus I and II, i.p. standard derivatives and antiderivatives, and integration techniques. If there is a defect on Precalculus techniques or College Algebra, then hurry up to back up this knowledge too. Do you know how to simplify e^{a + ln (b)} ? Are you able to simplify a/b/c/d? What is a limit? What is an improper integral? What is a function? What do you know about continuity, monotonicity and derivatives? Can you remember the monotonic sequence theorem from Calculus? Do you know how to find int^t s^n e^s ds for all integer n? Do you know the difference between int^t K(t,s) g(s) ds and int^t K(t,t) g(t) dt? Are you sure with integration method of partial fractions? Do you know partial derivatives and potential functions? Can you remember Taylor's formula and Mean Value Theorems of differential and integral calculus? What about trig formulas? What about solving linear systems of algebraic equations with respect to unknown variables? Check your knowledge! or Redo Calculus, Trig & Algebra, the sooner the better.

Remarks for the 1st Midterm Exam:

Please, review your notes and sections 1.1. - 3.2. The midterm exams are not of multiple choice. Midterm I will contain 5 problems (no word problems). The main goal is to check your capability of learnt ODE techniques. In particular, you should be sure in separation of variables, methods of integrating factor and variation of parameters, Bernoulli equations, homogeneous equations, exact equations and integrating factors, asymptotic stability and equilibrium solutions, Euler method, 2n order equations with constant coefficients, characteristic equations, fundamental set and general solution, and the role of Wronskian determinant. No (text)books, no notes, no tables, no graphic calculators at all, no notebooks are permitted (only plain calculators, i.e. non-programmable and non-graphical ones with the standard function evaluations). I don't expect that you know all formulas, but you should be able to understand and work with them. Go over the homeworks again to have a glimpse on those problems you may expect on the midterm.

Remarks for the 2nd Midterm Exam:

Please, review your notes and sections 2.1. - 4.4. The midterm exams are not of multiple choice. Midterm II will contain 5 problems. The main goal is to check your capability of learnt ODE techniques. In particular, you should be sure in the concepts of the Wronskian, fundamental solutions, Abel's representation, linear independence, linear superposition principle, characteristic equation, characteristic polynomial, roots of polynomials, complex numbers, Euler's formula, D'Alemberts reduction method, Euler equations, Bernoulli equations, method of variation of parameters, second and higher order equations with constant coefficients. No (text)books, no notes, no tables, no cheating sheets, no graphical calculators, no notebooks are permitted (only plain calculators, i.e. non-programmable and non-graphical ones with the standard function evaluations). You should be able to understand, apply and work with the formulas you have seen in class, your notes and / or in the textbook. Go over the homeworks again to have a glimpse on those problems you may expect on the midterm.

Remarks for the 3rd Midterm Exam:

Please, review your notes and sections 2.1. - 10.5. The midterm exams are not of multiple choice. Midterm III will contain 5 problems. The main goal is to check your capability of learnt second order ODE and PDE techniques. We concentrate on chapters 5 and 10. In particular, you should be sure in series solutions at ordinary points, boundary value problems for 2nd order ODEs, Fourier series computations, absolute convergence, power series, Dirichlet's rule, L^2-convergence, Parseval's identity, Bessel's identity, heat conduction problems, wave propagation equations, mixed problems, separation of variables and related issues. Thus, second order linear ODEs and second order linear PDEs are subjects in form of initial and boundary value problems. No (text)books, no notes, no tables, no cheating sheets, no graphical calculators, no notebooks are permitted (only plain calculators, i.e. non-programmable and non-graphical ones with the standard function evaluations). You should be able to understand, apply and work with the formulas you have seen in class, your notes and / or in the textbook. Go over the homeworks again to have a glimpse on those problems you may expect on the midterm.

Remarks for the Final Exam:

The written final exam has 10 problems, 10 points each, and lasts two hours. No textbooks, no calculators, no notes are permitted. We ask for 1st order equations (separation of variables, substitutions, Bernoulli equations, asymptotic stability), 2nd order equations (constant coefficients, characteristic equation, Euler equations, series solutions, boundary value problems), and 2nd order PDEs (Heat and Wave equations, Fourier analysis, separation of variables) with homogeneous boundary conditions. However, if the student agrees, then the final exam can be chosen as an oral one. In this case we would have some flexibility with your exam date and you would have one hour to demonstrate in an academic discussion that you are capable of the presented course material. You start with reporting on basic facts and topics of your choice related to the initial value problems. You may also prepare a presentation of more sophisticated topics from ODEs and PDEs if you wish to do so. It will be supposed that you use the blackboard and / or sheets of paper during this discussion. The final exam and the total grade will be determined immediately after that discussion too, unless there are further requirements to complete your course work (which I hope not). Please, contact me before the last week of teaching to set an exam time and date with me. The exam will take place with high probability in my office Neckers 265 during the last week and / or the exam week, based on our mutual agreement with class participants. Of course, if you still insist on then there will be a written final exam, but I honestly hope not. Don't have any serious fears towards the oral final exam. Mostly, we enjoy, both you and the instructor, because it aims to help you to improve your capacity in oral presentations.

Current Course Outline (Last Update: 04/17/03):

Week 1 : Introduction, 1.1.-1.4.
Week 2 : 2.1.-2.2.
Week 3 : 2.2.-2.4.
Week 4 : 2.5.-2.6.
Week 5 : 2.7.-2.8., 3.1.
Week 6 : 3.2., 3.3, Exam I
Week 7 : 3.3.-3.5.
Week 8 : 3.6.-3.9.
Week 9 : Spring Break
Week 10 : 4.1.-4.3.
Week 11: 4.3.-4.4., Exam II
Week 12: 5.1.-5.3.
Week 13: 10.1.-10.2.
Week 14: 10.3.-10.4.
Week 15: 10.5., Review, Exam III
Week 16: 10.6., 10.7 & Review

Remarks for Homeworks, Quizzes and Recitations:

The homeworks and quizzes are periodically assigned in classes and are due by the next Monday thereafter. The recitation discussion takes place probably every Wednesday during class hours. The homework is meant to strengthen your knowledge in related areas touched in lectures. Thus, your active participation is required. We plan to count the best 15 homeworks out of all submitted ones. Late homeworks are accepted in case of well-documented reasons beyond your control, but certainly not with full credit if it were under your control. Random quizzes with no make-up possibility are given if the overall participation dramatically drops down. However, we normally plan to evaluate more on the basis of homeworks. The homeworks and quizzes play an essential role in forming your grade according to the presented grade distribution.

Homeworks (Graded Assignments) (Last update: 03/19/03):

Week 17: 10.5.: no. 10 | 10.6.: no. 1, 12, 15 | 10.7.: no. 5 (due on 05/02/03)

Week 16: 10.4.: no. 16, 17, 18 | 10.5.: no. 1, 5 (due on 04/28/03)

Week 15: 10.2.: no. 14, 17, 27 | 10.3.: no.: 5, 14 (due on 04/21/03)

Week 14: 5.3.: no. 11, 21, 26 | 10.1.: no. 14, 16 (due on 04/14/03)

Week 13: 5.1.: no. 24 | 5.2.: no. 13, 21 | 5.3.: no. 8, 10 (due on 04/07/03)

Week 12: 4.3.: no. 2, 21 | 4.4.: no. 6 | 5.1.: no. 8, 18 (due on 03/31/03)

Week 11: 4.1.: no. 10, 16, 27 | 4.2.: no. 10, 24 (due on 03/24/03)

Week 10: 3.6.: no. 8 | 3.7.: no. 12, 27 | 3.8.: no. 4, 6 , 12 (due on 03/17/03)

Week 9: Spring Break

Week 8: 3.4.: no. 22, 40 | 3.5.: 14, 31, 40 (due on 03/07/03)

Week 7: 3.2.: no. 28 | 3.3.: no. 3, 8, 9, 17 (due on 03/03/03)

Week 6: 3.1.: no. 2, 22, 28, 39 | 3.2.: no. 5 (due on 02/24/03)

Week 5: 2.5.: no. 26 | 2.6.: no. 5, 18, 32 | 2.7.: no. 20 (due on 02/17/03)

Week 4: 2.5.: no. 3, 7, 14, 25 | 2.6.: no. 20 (due on 02/10/03)

Week 3: 2.3.: no. 3, 7 | 2.4.: no. 15, 16, 28 (due on 02/03/03)

Week 2: 2.1.: no. 16, 29, 37 | 2.2.: no. 20, 29 (due on 01/27/03)

Week 1: 1.2.: no. 7, 1.3.: no. 6, 8, 14, 25 (due on 01/22/03)

Further Readings : (optional)

• H. Amann: Ordinary Differential Equations, deGruyter, NY, 1983.

• V. Arnold: Ordinary Differential Equations, Springer, NY, 1980.

• M. Braun: Differential Equations and Their Applications, Springer, NY, 1975.

• E.A. Coddington and N. Levinson: Theory of Ordinary Differential Equations, MacGraw-Hill, NY, 1955 (Standard Reference).

• P. Hartman: Ordinary Differential Equations, Wiley, NY, 1964.

• H. Heuser: Ordinary Differential Equations, Teubner, Stuttgart, 1991.

• E. Kamke: Differential Equations. Solution Methods and Solutions, Teubner, Stuttgart, 1983.

• W. Walter: Ordinary Differential Equations (6th edition), Springer, NY, 1996 (Our First Recommendation).