MATH 231, FALL 2010-A. Freire-COURSE LOG

Th 8/19 Course policies
              First-order initial-value problems: linear vs. nonlinear, general existence-uniqueness theorem/ concept of implicit solution
              Problems (1.2): 10,11,16/24,27,28/31

Tu 8/24   Separable equations
               Problems (2.2) 7,9,13 (general solution)/ 17,21,25 (include the interval where the solution is defined)/31

Th 8/26  Linear equations
               Problems (2.3) 7,11,13,19/31,33 (include the interval of definition in all cases).

Course project- details
This assignment consists of writing a short paper (4-5 pages) describing an  application of material found in the text
(whether the material is due to be covered in lecture or not) to a problem in an area of science or engineering
of interest to the student; typically this is found in a text for a more advanced class in the student's major.
Rules: 1- The application in question may not be one already described in the text (or in a different differential equations text).
For examples of what is meant by "describing an application", see the "group projects" A-G in Ch.3 of the text.
2- Sources: written sources must be textbooks or original papers in the area of application. Internet sources are excluded.
3-Plagiarism (copying large chunks of material without attribution) will be detected and punished (zero on the assignment, treated
as academic dishonesty).
4-Human sources: by all means, ask a more advanced student or a professor in your major department (or intended major) for help
with this. Part of the goal of the assignment is to have them help you understand why this course is required. The names of people
who provided assistance must be included in the paper.
5-Structure- the paper should  have the following sections:
                   1-Introduction: statement of the general problem and of its importance in the applied area;
                   2- Formulation of a specific instance of the problem (numerical example)
                   3- Solution of the problem in part (2), based on material included in the text for this course
                   4- Conclusion (interpretation of the result, in terms of the science/engineering application considered).
                   5-References (texts/papers consulted, people providing assistance.)
6-Due dates: the project is due in two steps. On Oct. 5  you  must turn in a draft, including at least the first section (description
of the problem). I will review this and make comments. If this draft is not turned in on this date-no need to go on (zero on the assignment).
The final paper is due on the last day of class, Nov.30.

Tu 8/31 Exact equations
              Problems (2.4) 9, 11, 15, 17/ 21, 23 (explicit solution when possible, including the interval of definition).

Th 9/2  Special techniques: integrating factor, Homogeneous type, Bernouilli equations, dependence on ax+by, linear differential forms
            Problems: (2.4) 29, (2.6) 9, 11, 17, 19, 21, 23, 29, 31

Tu 9/7  Applications. Problems  (3.2) 3, 5, 19, 21, 25 (3.3): 3, 7, 9, 13

Th   9/9 Applications: mechanics. Problems: (3.2) 15, (3.4) 1, 9, 13, 25

Please note:  office hours will be  on Tuesdays, 5-5:45 Thursdays 12:30-1:30, or by appt (just before the problem session).

Tu 9/14  Geometric analysis of autonomous equations

Th 9/16 EXAM 1: Lectures from 8/19 to 9/14 (section 3.5 not included)
Exam 1-Problems
Exam 1-solutions

Tu 9/21 Linear second-order equations (homogeneous); simple harmonic motion, phase-amplitude form; examples with real roots
Problems: (4.2) 13, 15, 17, 19

Th 9/23: examples with complex roots (REVIEW COMPLEX NUMBERS PRIOR TO LECTURE)
Problems (4.3) 9,13,17,21,23,25

Tu 9/28; non-homogeneous equations
Problems (4.4) 11,12,13,15,16,19,20,23
non-homogeneous equations-examples

Th  9/30: non-homogeneous equations--superposition, variation of parameters/ non-constant coefficients: existence/uniqueness theorem,
Cauchy-Euler equations
Problems: (4.5): 13, 14, 20, 21 (find the general solution) (4.6): 5,9,11, 17 (4.7)1,5,7,9,11,15,21

Tu 10/5:  Cauchy-Euler (complex roots), reduction of order, qualitative aspects (conservative autonomous equations)
nonlinear second-order equations: qualitative aspects
Problems (4.7) 37, 39, 45, 48


Tu 10/12: oscillations in mechanics: periodic external force, resonance
Problems: (4.9): 3, 5, 19, 13, 18 (4.10): 1,3,5,9,15

Th  10/14: systems (5.1, 5.2): solution by substitution (non-degenerate cases)
Problems (5.2):5,7,9,23,29,31,35
Remarks on the course project:
1- either the DE considered should be non-standard (not discussed in class) or, if the equation is standard, the applied model
should not be a standard example, and a detailed derivation of the DE should be included.
2-at east one solved numerical example should be included, including appropriate graphs and interpretation of the result (in terms of the application).
3-NO COPYING! If all your material is from one source, make sure you understand it and re-phrase the explanation in
your own words.
4-The structure described above (under "rules")  must be adhered to.

Tu 10/19: systems: degenerate cases, applications to coupled oscillators. Problems (5.6): 1,3 (use all masses=1 and all spring constants
equal to k)


Th 10/21: EXAM 2. Material: lectures from 9/21 to 10/19, online handouts
Exam 2

Tu 10/26 Laplace transforms (Ch.7)-definition, property, basic transforms
(7.2) 11, 17, 29, 30 (7.3) 5,7,9,11,15,35

Th 10/28 inverse transforms, solution of IVPs
(7.4) 5,7,9,21,23,33,41 (7.5)1,3,5,29,35

Tu 11/2 Laplace transforms: discontinuous and periodic functions/systems
(7.6) 3,7,9,11,13,23,27,37 (7.9) 1,5,13

Th 11/4 Laplace transforms: convolution, Dirac delta function
(7.7) 3,23,35  (7.8)1,3,5,13,15,21

Tu 11/9 and Th 11/11 Analytic functions, regular points, solutions by power series
(8.2) 1,3,5,29,31,33 (8.3)11,15,17, 21 (8.4) 1,3,5,7,9

Tu 11/9 Problem session (Laplace transforms) Tu 11/16 Problem session (power series solutions)

Tu 11/16, Th 11/18 : solutions by power series for second-order DEs with singular terms (Frobenius method)
Problems: (8.6) 1,3,5,11,13,15, 19, 21 (8.7) 1,3

Tu 11/23 EXAM 3. Included: lectures from 10/19 to 11/18, online handouts
Exam 3

Th 11/25 Thanksgiving

Tu 11/30 solution of exam 3, course project due, course evaluations

FINAL EXAM: Tuesday 12/7, 8:00-10:00AM
The final will consist of 8 problems, taken from exams 1,2, and 3 (with slight changes)