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