MATHEMATICS 231- DIFFERENTIAL EQUATIONS- SPRING 2008- COURSE LOG

Th  1/10       Course policies/ different types of differential equations/ linear 1st order equations (temperature example)

Tu  1/15       linear 1st order equations: concentration example. Non-constant forcing term: periodic example.
Complex numbers: the deMoivre-Laplace formula, computation of integrals. Phase-amplitude form
Homework set 1
(due date: Thursday, 1/24)

Th  1/17       First-order linear equations with variable coefficients: examples, existence/uniqueness theorem for
the IVP ( main example: y'+p(t)y=0 with p(t)=2/(t^2-1)-diagram of all solutions.)

Tu   1/22      First-order linear equations.
First-order linear equations- examples (summary)
(for practice, try to solve the examples independently-answers are given-and sketch their graphs)

Th 1/24       First-order linear equations with periodic coefficients: four examples
Second-order homogeneous equations, const. coeff. (start): hyperbolic sine (sinh) and cosine (cosh).
Homework set 2
(due date: Thursday, 1/31)

Tu  1/29       Second-order homogeneous equations: simple harmonic motion and
motion under repelling force proportional to distance.  Conserved energy (in both cases).
Oscillatory, bounded solutions vs. unbounded solutions. Representation in (position, velocity) graph

Th  1/31       Second-order homogeneous equations with damping term- 2 of 3 cases (oscillatory underdamped, overdamped stable)
Graphs of solutions and representations in (y,v) plane. Parameter space (b,c) for y''+by'+cy=0.
Discussion of Hw 1
Homework set 3
(due date: Thursday, 2/7)

Tu  2/5      3rd case: unstable motion under repelling force witth damping term: general solution, (y,v) graphs/ non-homog eqns (start)

Th    2/7      non-homog 2nd order eqns: periodic external force, resonance.
Homework set 4
(Will be discussed in class on 2/12; problems turned in at the beginning of class will count towards the
homework grade- but make a copy of what you turn in, so you can study for the test.)

Tu 2/12      review session (problems from Hw sets 2,3,4)

Th    2/14   Exam 1

Tu  2/19     first-order systems: reduction to std form, equivalence w/ 2nd order eqns, sln by substitution/ first-order matrix systems:
defn of eigenvalue-eigenspace, characteristic polynomial, connection with 1st order matrix DE/ brief discussion of Ex.1

Th  2/21     first-order systems: saddles, stable/unstable nodes: eigenspaces, general solution, graphs of solutions, conserved energies
complex eigenvalues:  complex general solution and real-valued general solution
Homework set 5
(due Thursday, 2/28)

Tu  2/26     complex eigenvalues: diagram of all solutions- stable/unstable spirals.  Trace-determinant diagram, stability under small
perturbations. Special cases: centers, zero eigenvalues, equal eigenvalues (lecture in computer lab- MATLAB demonstration.)

MATLAB function m-files dfield7 and pplane7 are found here:
dfield7 (by John Polking, Rice U.)

Th  2/28      The fundamental matrix of a 2X2 linear system. Application: solution of non-homogeneous systems using the
(matrix) variation of parameters formula.
Homework set 6  (due Thursday, 3/6)

Tu  3/4       Coupled oscillators: springs with the same constant, normal frequencies./ solution of Hw set 5.

Th 3/6       Solution of Hw set 6

Tu  3/11    Laplace transforms:  first properties, use to solve 2nd order DEs, computation of the inverse transform

Th 3/13     Laplace transforms: discontinuous forcing terms (Heaviside step function), convolution product, systems (example).
Homework set 7
(due 3/27)

3/18, 3/20: Spring Break (no classes)

3/25          Nonlinear autonomous equations (first order): graphical analysis. Stable/unstable equilibria, blow-up in finite time,
graph of all solutions. Two conditions for global existence. Eqns of Bernouilli type.
Graphical analysis of autonomous equations

3/27:         non-linear autonomous equations (first order):  Riccatti equations, shifting property. Breakdown of uniqueness (example)
The E/U theorem.  The catenary problem.
Examples with Riccatti equations (ignore the references to `problem 3' and `problem 4' at the beginning)
Homework set 8- nonlinear autonomous equations (due Tuesday, 4/1)

4/1:             Review class (solution of Hw sets 7 and 8)

4/3:            Exam 2

4/8:             Exam 2 returned. Second-order autonomous conservative equations: qualitative analysis based on graph of the potential energy
Handout:  second-order equations and mechanics
(includes homework problems)

4/10          Newton's derivation of Kepler's laws
handout 1: Newton's derivation of Kepler's laws
handout 2: From elliptical orbits to the inverse-square law
(includes homework problems)
Homework set 9 (due 4/17): problems in handout 2 of 4/10 ; problem 2 in Exam 2 of Fall 2007: here

4/15       elliptical orbits imply an inverse-square law (inc. problem 2 from handout 2 above)
exact equations
(includes 6 problems, also due as part of HW 9 on 4/17: try to turn in at least 3, at least 1 from each of part(A) and part(B)

4/17        first-order equations of `homogeneous type' (invariant under dilation)
second-order linear equations with variable coefficients:  existence-uniqueness theorem/ Liouville transformation/
reduction of order/ solution of non-homogeneous equations by `variation of parameters'
Homework set 10
(due 4/24)

4/22       examples related to the material introduced 4/17

4/24      comments on Hw sets 9 and 10

FINAL EXAM : 10:15-12:15, Thursday, May 1. Review: Homework sets 1-10, Exams 1 and 2

Final Exam