MATH 231 FALL 2007- COURSE LOG

Th, 8/23        Course policies/ exponential function
HANDOUT 1 (PDF)
(includes HW problems-due Aug. 30)

Tu, 8/28        Exponential models
Examples solved in class
Exponential models- HW problems
(due Sept 6)

Th  8/30        Exponential models (remaining examples)
discussion of  HW problems 3 and 4, handout 1
general facts: autonomous equations, the geometric meaning of the E/U theorem
summary of discussion- handout 2
(includes 6 HW problems, due Sept. 11)

Tu  9/4         general linear first-order equations (sect 1.2 in [Braun])-first examples
(steady-state slns; amplitude and phase shift for periodic functions)
examples solved in class (linear equations)
linear equations- HW problems
(due Sept 13)

Th 9/6          1st order equations w/ non-constant coefficients (cont.)
discussion of HW problems

Tu 9/11         The existence-uniqueness theorem- statement, geometric interp.
separable equations- general solution, domain of existence, finite-time blowup
(contrast between linear and nonlinear eqns)
The E-U theorem and separable equations- HW problems
Due date:  Sept 20

Th 9/13        separable equations- final examples, homogeneous equations
graphical analysis of autonomous equations
(includes homework problems, due Sept. 25)

Tu 9/18         autonomous eqns. (cont.)
solution of HW problems
exact differential equations
(includes 6 homework problems, due Sept. 25)

Th 9/20         exact diff equations (end)
Bernouilli equations

Tu 9/25         discussion of HW problems

Th 9/27         Exam 1

Tu 10/2          Three applications of first-order equations
Exam 1 returned (inc. written solutions)

Policy  on oral exams.  Minimal requirements for an oral exam will be:
(i) attendance to every class beginning  on 10/4
(ii) turn in at least 6 complete homework problems per set (from now on), of which at least 4 correct
(iii)  solve all 20 problems in the following problem set on first-order equations (at least 80% correct):
Supplementary problems
Due date: 10/18
Remark:   these problems will not count towards the homework grade- they are meant as a review for
those students interested in taking an oral exam.

Th 10/4      Linear second-order equations (sect 2.1, 2.2 in [Braun])
E-U theorems in the linear and non-linear cases (contrast)/ homog. equations with constant
coefficients: general solutions (3 cases)/ method of  "reduction of order"

Tu  10/9       Non-homogeneous linear 2nd-order equations (const. coeffs.); De Moivre's formula
Non-homogeneous equations- examples
Homework problems-2nd order equations  (due date:  10/16)

Th  10/11      Fall break (no classes)

Tu 10/16       Non-homogeneous equations (examples)
Second-order equations and mechanics
(First version: includes only material presented on 10/16- continuation, including HW

Th  10/18        Simple harmonic motion-examples
Homework problems
(due date: 10/25)

Tu  10/23        Discussion of homework problems; method of variation of constants
Damped harmonic motion

Th   10/25      Forced oscillations/resonance
Damped forced oscillations/resonance frequency, amplification factor
Homework problems
(Source: Tenenbaum-Pollard; DUE DATE: 11/1)

Tu  10/30       Motion under a central force/
Newton's derivation of Kepler's laws
From elliptical orbits to the inverse-square law
(includes proposed problems)

Th   11/1         Power-series solutions of second-order equations
Homework problems
(due date: 11/8)

Tu    11/6       Power -series solutions (final examples)
Discussion of homework problems

Th    11/8       Discussion of HW problems/  solution of 2X2 1st order systems by substitution

EXAM 2: Tu 11/13, topics included: lectures from  10/2 to 11/6 (inc. problems and theory in 10/30 handouts)

Tu    11/13       Exam2

Th     11/15      Laplace transforms: basic properties; Heaviside's formula
Homework problems
(due date: 11/27)

Tu     11/20      Laplace transforms: convolution formula, transfer function/impulse response function, solution of systems
Exam 2 returned (inc. written solutions)

Th   11/22        Thanksgiving holiday

Tu    11/27      The heat equation and the wave equation/ eigenvalue problems in an interval/
example (initial-boundary value problem for the heat equation)
Homework problems
(due date: 12/4)

Th      11/29      Example: wave equation (Neumann) BC. Infinite series of eigenfunctions: Fourier series,
Fourier sine and cosine series/ Solution of PDE using Fourier series (example: wave eqn, Dirichlet BC)/
Example: an eigenvalue problem w/ eigenvalues found `graphically'.
Course evaluations filled out.

Tu     12/4         Discussion of HW problems (last 2 sets)

FINAL EXAM: Tuesday, Dec.11, 12:30-2:30 (in the usual classroom)

Topics: the exam will consist of six problems: 2 based on problems in Exam 1, 2 based on problems in Exam 2
and 2 based on the material taught in the lectures  from 11/8 (inclusive) to 12/4, especially the last two homework sets
(posted 11/15 and 11/27).

Final exam