MATH 231 FALL 2008- COURSE LOG

8/21   Th         Course policies
                        First-order DEs: general  properties: interval of existence, implicit solutions,
                        one-parameter families. Direction fields and integral curves
                        homework  (4 problems, due 8/28)

8/26   Tu          Regular points and singular points in the (x,y)-plane for a 1st order DE.
                        The main theorem for equations of the form y'=f(x,y), y=y(x). First-order equations
                        stated using differentials: P(x,y)dx+Q(x,y)dy=0  (only those (x,y) for which P(x,y)=Q(x,y)=0 are singular)
                        Separable equations: examples, graphs, domain of definition. ([TP  lesson 6C])
                        homework (due 8/28)

8/28    Th          Equations of `homogeneous' type (reduction to separable by substitution)
                         homework (due 9/4)
                         answers to a student's questions


9/2        Tu        example: eqn of homogeneous type (airplane)-p.168)/general discussion of 1st order eqns
                          (existence/uniqueness, domains: linear vs. nonlinear eqns, finite time blowup/implicit function theorem)
                          homework problems: lesson 17A, numbers 4 and 6  (p.176)- due 9/4 (accepted also 9/11)


9/4         Th        example: the suspension cable
                          Hw:  read example 36.36, p.511; do problem 2, p.515 [due 9/11]
                          exact differentials/solution curves and smooth level sets/ criterion for exactness/ "partial integration" method
                           Hw:  p.79- 4,6,8,16 [due 9/11]

9/9         Tu        example: closed differentials which are not exact in the whole plane, but are exact in other regions (primitives based on arctan)
                           integrating factors/ discussion of homework
                           homework (due 9/18)

9/11        Th         linear equations (existence/uniqueness thm, domains) Bernouilli  eqns, Riccatti  eqns
                             Homework (due 9/18) : p.97:  6, 7, 15,17 (use the IC: y(pi)=1, include domain), 21 (include domain), 29

9/16         Tu        Some applications of  constant-coeficient linear equations
                              graphical analysis of autonomous equations  (to be discussed Tu+Th- read before class-revised 0917/08)
                             Homework (due 9/25):   p.124:  9,11  p.130: 4 p.133: 7,8         above handout: problems 3,4,5

9/18        Th         handout on autonomous equations/ autonomous Ricatti equations/ random examples from p.103/104
                              autonomous Riccatti equations (handout, includes 2 problems)

Studying for the test:  make sure you know how to solve all the problems assigned as homework, and also the problems in
this last handout (posted 9/18). Try and solve a random selection of the problems on p.103/104 of the text. You
probably know this already, but the only way to do well on a math test is to get practice by solving lots of problems
correctly (it can't be done if you start the night before the test.) On Tuesday we'll have a review session, driven entirely
by questions from students.

 1st. test: Thursday, 9/25  Material included:  lecture notes, online handouts, sections in text from which HW was assigned. (until 9/18)
                         
 9/23   Tu     Review
9/25    Th     Exam 1 (24 present)

9/30    Tu     Second order linear equations: Linearly independent solutions, Wronskian, form of the general solution.
                   Constant coeff equations: case of real distinct roots/  solution of Exam 1
                    Homework:  p. 220: 1, 3, 31  (due 10/7)  also: review complex numbers (lesson 18, p.197) if needed (will be used on Thursday)   

10/2     Th   Const coeff eqns: homogeneous w/ complex roots (real form, phase-amplitude form), with repeated root.
                    Non-homog equations: method of undetermined coefficients (use of operator notation)
                    non-homogeneous equations- examples (will be discussed in class on Tuesday 10/7)
                    Homework: p.231: 4,9,13 (due 10/7)

10/7     Tu   Non-homogeneous second-order linear equations: solution by `undetermined coefficients' (using complex numbers)
                    Homework (due 10/16):  p.231/232:  7, 8, 15, 16, 17, 22, 24, 29

10/9   Th   Fall break (no classes)

10/14  Tu      Non-homogeneous eqns- solution by variation of parameters/ simple harmonic motion; examples
                      
10/16  Th       damped free oscillations/ motion under periodic force with and without damping/ Energy, equivalent
                        first-order systems, phase-plane diagram of solutions
                    Homework: due 10/23: pick a total of 4 problems from 28 A/B, 28C, 29A, 29B (one from each group)
                          Rule: 0.5 point for a standard "plug in the numbers"-type problem; 1.5 point for an "interesting" problem .
                           (Interesting problems are generally found twds the end of each group, but I'll have to agree with you that it
                              is "interesting")

10/21  Tu        forced motion with damping- amplification factor, resonance frequency, condition for resonance/ 2x2 systems:
                       solution by substitution, using differential operators./ criterion for the number of arbitrary constants
                    Homework (due 10/30):  p.421:  2,5,8,9,12,15

10/23  Th      Homogeneous first -order systems via "matrix theory": (2X2 matrices) : eigenvalues/eigenspaces, general solution in
                      vector form, phase plane diagrams. Complex eigenvalues: real-valued general solution.  Types: saddles,
                      stable/unstable node, stable/unstable spiral, center
                       Problems on 2x2 first-order systems of DE
                        (solve these problems for practice- this is the kind of problem on this material that may appear on the test)

 Exam 2: Thursday 10/30. Topics: lectures and Hw from  9/30 to 1/0/23:  2nd order equations and applications, systems (constant coefficients,
                      homogeneous or not)
Exam 2 
(17 present)
11/4 Tu         Non-linear second-order equations: special types y''=f(y) (conservative mechanical systems) y''=f(t,y'), y''=f(y',y)/
                   solution of exam 2. HW problems:  1,2,3,6,8,9,18,19 (p.504)   due date 11/13

Optional exam 3: Tuesday 11/25, 6PM (alt: 11/26, 6PM)

11/6  Th          Conservative mechanics- see handout: 2nd order equations and mechanics
                    The Kepler system- see handout (this is a reading assignment): Newton's derivation of Kepler's laws
                    In text: lesson 34. Problems (due 11/20): 1 to 6, p. 475

11/11  Tu       Derivation of the inverse-square law from elliptical motion: From elliptical motion to the inverse-square law
                        (includes HW problems, due  11/20)

11/13  Th      Reduction of order, Liouville transformation, power-series solutions (start)
                      Problems on power series slns (due 11/25): p546- 5, 6,10, 11  Reduction of order: p.246: 9, 12, 14
                      From now until the end, all problems turned in will be graded (i.e., no longer limited to 4)

11/18  Tu       Power series solutions/Laplace transforms
                       Problems on Laplace transforms (due 11/25): 16, 18, 21 (p.311)

11/20  Th       Laplace transforms: discontinuous functions, periodic functions, convolution theorem
                       Practice problems on Laplace transforms
                       Material included in next week's test: lectures from 11/4 to 11/20 and corresponding problems

11/25  Tu      Review- Exam 3   Exam 3B
                      Solutions

11/27 Th   Thanksgiving  (no classes)

12/2   Tu       The brachistochrone problem     (reading for the Thanksgiving break)

FINAL EXAM:  Thursday, 12/11, 12:30-2:30 (in Ayres 214, as usual)
Comprehensive: for the material included in exams 1 and 2, the questions will be based on those in the corresponding
exams. For the material in Exam3, questions may be based either on these tests (3/3B) or on the problems
(from text and handouts) assigned as HW 11/4 to 11/20. (The material in the handout of 12/2 is not included).

Final Exam
(15 students present)