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
Th 10/7-FALL BREAK
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
PROBLEM SESSION 5:45-7:00
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)
PROBLEM SESSION 5:45-7:00
Th 10/21: EXAM 2. Material:
lectures from 9/21 to 10/19, online handouts
Exam
2
solutions
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
solutions
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)