MATH 231- SPRING 2009- COURSE LOG

Th  1/8     Course policies /First-order equations: standard form, differential form, general remarks
Separable equations: examples, solutions in explicit form and domains
HW (due 1/15):  read 6C/ p.55 4, 7, 12/p.55 19, 20  (include the solution in explicit form, and its domain)

Tu  1/13   regular and singular points for 1st order equations (standard form, differential form)/ 1st order
equations of homogeneous type (7A/7B)
HW (due 1/15): p.61: 5,6,10 (in each case, identify the regular points); 12,15 (the answer should be the explicit solution y(x),
including its domain- this will be different from the answer in the book.)

Remark: although 10 problems are designated "due 1/15", turn in at most 5 for grading (you may , of course, work on all 10 for practice).
In general, it is a good idea to pick your problems from more than one section- say, 3 from those listed on 1/8 and 2 from 1/13.

ODE plots with Matlab

Playing with this is completely optional- however,  you'll find that generating graphics effortlessly makes
the material a lot more entertaining and easier to understand.  Although all engineering majors acquire expertise
with Matlab at some point, you don't need to know any Matlab to use this- it is all "point and click".   The
exception is that you need to know how to open an m-file editor, just to copy the code
from the web site to your computer. I'm working on having these m-files installed campus-wide, but in the meantime you could ask a friend
for help with this one step. If enough people are interested, I may teach an (optional) after-hours Matlab session in
a few weeks. Oh yes, if you turn in a relevant Matlab plot with a homework problem, you'll get "good karma" recorded for that problem.

Th 1/15  Comments on HW problems/Pursuit problems (17A), exact DEs (9)
HW (due 1/22) p. 175: 1,2,6
p. 79 5,6,8 (for these 3 problems, find also the singular points and corresponding values of c) 16, 17
Tu 1/20    Exact equations (cont.)- finding a primitive, integrating factors
HW (due 1/22)  p.90: 3,8,12- you may use the integrating factor given in the answer without explaining how it is found.  Discuss if
there are solutions of the original problem not captured by the general solution of the modified problem.

Enrollment (add/drop closed): section 3=22 /section 5=34

Additions to and clarifications on the homework policy:
a)  Homework must be turned in at the beginning of class; if you arrive late, come to the lectern and deposit your homework silently
before taking your seat. Homework will NOT be accepted at the end of class (and, of course, not later.)
b)  The book includes answers to all problems, and you MUST continue the solution until you match the answer in the book (or, if it is
wrong, explain why.) This will not really be an exercise in DE, but rather on manipulations involving algebra and trigonometry identities,
which is good (and needed) practice.
c)  The list of problems given in this "course log" will sometimes include additional questions not asked in the book. These questions MUST be answered,
or you won't get credit for the problem (even if your answer matches that given in the book.)
d)   any problem given as HW is considered "fair game" for a test; even if you don't turn it in, or if I don't solve it in class, it is the student's responsibility to
make sure he/she can solve it (make an appointment to come to my office if you can't solve a problem after trying for an hour or so).

Th 1/22 Linear and Bernouilli equations (Lesson 11) /discussion of HW problems
HW (due 1/29) p.97: 3,6,7,21,23 (for the last two problems, find the solution explicitly and include its domain)

Tu  1/27 Miscellaneous methods (lesson 12)/Applications of 1st order equations (15A,C,D,16A)
HW: p. 103: 3,11 (for practice, try some of the others not done in class as well)
HW: p. 124: 11, p.130: 4, p.133: 6,8, p.153: 16(a)
(these problems are due 1/29)

Revised office hours  (for the remainder of the spring term): Tu+Th, 12:45-1:45 (Aconda 406B) or by appointment (e-mail or 974-4313)
Remark: typically I don't read e-mail after 7PM, or before 9AM

Th  1/29   1st order nonlinear autonomous equations (15 E and the online handouts below)
Graphical analysis of autonomous equations
Autonomous Riccatti equations
Problems on this material:  both handouts include problems. In addition, from the text:, read examples 15.52, 16.26, 16.27

Tu  2/ 3   Discussion of problems from 3rd HW set and handouts

1st test: Thursday 2/5  Included: lectures up to Th 1/29 and associated problems.
Studying for the test:  the main thing is solving as many problems as you can on your own (especially those assigned as homework),
without looking at the examples in the text or class notes. The list of equations on p. 104 is a good source of practice problems. If
you just read worked-out examples without solving problems independently, you will fail the test.

Th   2/5   Exam 1
attending regularly: Section 3=11 (10 took exam 1)/ Section 5=22 (21 took exam 1)

Tu    2/10    Linear second order equations: constant coefficients/complex exponentials/characteristic equation/
Solution of exam 1
uniqueness for second-order linear equations  (optional reading)

NEW policy:  QUIZZES - For the next few weeks there will be 5-min quizzes (1 problem) at the end  of the lecture
(when time allows), dealing with material from the first part of the course (first-order equations). I will list in advance
groups of problems from the text (see below), and any of the problems from the list (or a problem similar to it) may
occur on the quiz. After 8 quizzes, I will total the score (over 100) and take the average with  Exam 1, if the quiz grade is higher
than the grade on Exam 1; this average will be your new exam 1 score. Note there is no penalty for not taking the quizzes

Th  2/12    Const coeff. 2nd order equations: complex roots, solution in real form and in phase-amplitude form  (Lesson 20- second order
only; review Lesson 18 on complex numbers, if needed.)/ The parameter space- oscillatory vs. non-oscillatory solutions
QUIZ 1: Exercise 6 (all), pp. 55-6 (if an IC is given, try to find the explicit form of the solution, including its domain.)
HW: p.220-6,26,32 (due 2/19)

Tu   2/17   Non-homogeneous equations: undetermined coefficients/ physical interpretation of parameters/steady-state solutions, energy conservation
(Lessons 21, 22B). HW-p.231: 6, 8, 21,31 (due 2/19)
QUIZ 2: Exercise 7, p.61 (all)

Th  2/19    non-homogeneous equations: undetermined coefficients (cont'd)/ variation of parameters (lesson 22)/
HW 5 (p.240)- 4,   9, 16, 18 (due 2/26)
non-homogeneous equations: examples
QUIZ 3: Exercise 9, p.79 (all)

Tu   2/24:  some applications of constant coefficient equations- sections 28 ABC (SHM), 28D (forced HM), 29A (damped HM), 29B (forced damped motion)
except for section 29B, no `theory' will be presented in class- these sections are a reading assignment. On 2/24, I will solve
a few representative examples in class.

HW 5 (due 2/26)- p. 321: 8, 12, 15 p.329: 7, 10, 12, 14, 21/p.336: 37/ p.343: 9/p.354: 12/p.364: 6, 7
QUIZ 4: Exercise 10, p.90 (use the integrating factors given in the answers)
New homework policy: starting with the HW set due 2/26, up to 8 problems will be graded. (The total of 40 pts for 100% HW grade
remains the same.)

Th 2/26:  continuation of sections from 2/24; discussion of the variation of parameters formula (read the handout below)
The variation of parameters formula
QUIZ 5: Exercise 11, p.97(all)
HW6 (due 3/5): p.364:6,7 (if not already done) 8, 12, 13, 14, 15;/p. 329: 4, 20, 21, 25, 28

Tu  3/3      systems of first-order equations (31D,E,F)- solution by substitution/ QUIZ 6: Exercise 15A, p.124
HW6 (due 3/5): p.421: 7, 8, 9, 12, 17, 18, 20/ Remark: give your answer in `vector form', and answer the questions:
(i) what is the asymptotic behavior of solutions as t tends to +infinity? (ii) do all IVPs have a solution?

Examples of linear systems
(These are the examples discussed in class, but the notes include aspects of the discussion-vector/matrix notation, degenerate vs.
non-degenerate systems, asymptotic behavior-not addressed in the book.)

Th  3/5   Quiz 7: Exercise 15C, p.130/
non-homogeneous equations with discontinuous forcing term (using the var. of parameters formula)/
from second-order equations to 1st-order systems/ HW 6 due
Linear differential equations with discontinuous `forcing term'
(Includes 5 practice problems, to be discussed on 3/5; only the case of  first-order equations is included in Exam 2).

Tu  3/10  Quiz 8: problems in the online handouts on autonomous equations  (see the links for Jan. 29)
Discussion of  HW problems and of problems in the 3/5 handout.

Th  3/12  Exam 2 /REMARK: CALCULATORS WILL NOT BE ALLOWED ON THIS TEST (Neither will `formula sheets'.)
Material included: lectures from  2/10 to 3/10- online handouts and sections in the text.

Studying for the test: solve as many problems as you have time for, especially homework problems and problems in the
online handouts (without calculators!). Read the summary descriptions in this `course log' and ask yourself if you know (without looking at your
notes or the text) what the main concepts/problems for this material are.  For extra practice, look at  "Exam 2" for fall 2008
(link included in the title page for this course, Math 231 Spring 2009.)   See also the corresponding  exams for fall 2007 and spring 2008
(links under `past teaching' on my web page), although the material included may differ slightly.
(If all you do is read solutions worked out in class or in the text, without attempting problems independently, you will definitely fail.attending regularly: Section 3=8 (7 took Exam 2)/ Section 5=19 (17 took  Exam 2)

Tu 3/17  and Th 3/19:  SPRING BREAK

Tu 3/24 and Th 3/26: Introduction to computer methods for ordinary differential equations. There are two handouts. (The first one has homework problems,
due 4/2- HW 7)
:
Introduction to computer methods    (9 pages,PDF)                   Error estimates for numerical methods
You are expected to learn the material  in the first handout.  The material in the second handout is more `advanced' (it has proofs!)
Other m-files you will need (eul, rk2 and rk4 were downloaded from John Polking's web page at Rice University).
It is probably a good idea to put all of these in a single folder (called, say, ode tools) along with dfield7 and pplane7 (see lecture of 1/13)
euler method       midpoint euler     Runge-Kutta 4    error analysis  instability

Tu 3/31     34 C,D,E: Motion under a central force: conservation of energy, all central force fields are conservative/ motion under force proportional
to distance/ planar motion in polar coordinates/ HW 7 (due 4/2): p. 476: 2, 4 (find the potential)/ p. 480: 6,7
Due to an unexpected "math server blackout" on 4/1,  HW7 may be turned in on  Tuesday 4/7.
Also, from this point on all HW problems turned in will be graded.

Th 4/2         Lesson 34 F,G,H:  from the inverse-square law to elliptic motion and conversely/ orbit parameters from the  initial conditions
Newton's derivation of Kepler's laws
(This handout includes some history of the problem)
From Kepler's laws to the inverse-square law
HW 8 (due 4/9):  p.488, no 2: use mass=1 kg, position =12 m from O, velocity=6 m/s, force=120/r^2
no 3: use mass=1 kg, position=10 m from O, force=100/r^2
p.497: 4,6,7,8

Tu  4/7       Nonlinear second order equations(lesson 35): reduction to a first order system, special cases; autonomous conservative
(analysis based on the graph  of the potential), general autonomous (reduction to a first order equation)
Autonomous conservative second-order equations
HW 8:  p.504:  1, 11 (r(0)=1, r'(0)=1), 14 (y(0)=0,y'(0)=1), 18,19 (due 4/9)
(i) where initial conditions are given, find the solution explicitly, including its domain.
(ii) if the eqn is autonomous conservative, include a graph of the potential , find/classify the equilibria
and include the range of the solution with the given IC.
Extra credit: students including a relevant MATLAB plot with the solution (either a y vs. t graph or a (y,y') graph) get *one* additional  point
for that problem
Example of a MATLAB m-file for a non-linear system
(this page may be copied directly as an m-file and saved to the folder containing the other m-files for this course)

Th  4/9     Laplace transforms (start) : Lesson 27
Practice problems on Laplace transforms
Remark:  Solve these problems as practice for Exam 3; those problems not solved in class on 4/14 may be turned in (as HW9)
on 4/23 (this means any problem on that handout, except 1(iii)).

Tu  4/14   Discussion of problems from HW 7 and HW8, and on Laplace transforms.

Exam 3 will be on Thursday, April 16.  Material included: lectures from  3/24 to 4/14 and corresponding online handouts and HW problems.
Calculators will not be allowed, but a table of Laplace transforms will be included.
The last HW set (HW 9) will be collected on 4/23 (Thursday)

Th 4/16             Exam 3     (section 3: 6 students/Section 5: 14 students)

Tu    4/21
Laplace transforms: the convolution theorem and applications/example with discountinuous forcing/ The catenary (suspension cable, 36A)
HW problems: p.514 2(a)(d), 3(a)(c), 4(a)(b)(c)
(these may be turned in as HW9 on 4/23, along with the problems in the Laplace transforms worksheet.)

Th    4/23  Matlab examples:  1st-order using dfield7 (including discontinuous forcing term); 2nd order using reduction to a 1st-order system and
pplane7 (when autonomous) and/or direct coding. The examples are found here:
Matlab examples (second order)
(may be copied and saved directly as an m-file)

FINAL EXAM:   Section 3: Thursday, April 30, 10:15-12:15 / Section 5:  Tuesday, May 5, 2:45-4:45 (in the usual classroom) Confirm here: Finals Calendar

Material included:  COMPREHENSIVE, first to last lecture.
The problems will be drawn from the following sources (possibly with slight changes):
(1) Lectures on 4/21 and 4/23, and the Laplace transforms worksheet
(2) The handout "introduction to computer methods" (lecture of 3/24)
(3) Exams 1,2, and 3 this semester (Spring 2009)- written answer keys have been given
(4)  Exams 1,2,3 or the final last semester (Fall 2008)-link given on the math 231 (spring 2009) title page.
(Of course, no need to look at the problems on material I did not present this time,  such as power series solutions)
Calculators and formula sheets will not be allowed on the final (A table of Laplace transforms and non-trivial integrals will be provided)

Final Exam