MATH 231 FALL 03

COURSE LOG

8/21-  linear  1st order equations
Ex.1 bacterial growth
Ex.2 growth w/ non-homogeneous term
Ex.3 temperature of body in a heat bath

8/26- linear equations
solving const coeff eqns by transformation; graphs
general non-homog eqn with vble coeffs: genl sln

8/28-linear equations (2.3, ch.3)
existence/uniqueness theorem; domain of def'n of solutions
example showing hypotheses are necessary
applications:  periodic source term ,  escape velocity

9/2- nonlinear first-order equations
examples: growth with negative quadratic term, free fall with
squared-velocity resistance.
main properties: trapped solutions, blowup in finite time
qualitative analysis of autonomous eqns: constant solutions, phase line
(see "group projects" in chapter 1, part D.)

9/4- nonlinear first-order eqns.
solution of hw set 1 (outline)
example: quadratic growth- qualitative analyiss, sln as a Bernouilli eqn,
blowup time, growth rates
inverse problem: find a model, given some qualitative features

9/9- autonomous first-order eqns: qualitative analysis
cubic example (Bernouilii), quadratic example (Riccatti); semi-stable example
conserved quantities and implicit general solutions (3 examples)

9/11-exact eqns: finding conserved quantitiies
integrating factors
local existence-uniqueness theorem; geometric interp.

9/16- 2nd order linear equations (ch. 4)- const. coeffs.
general solutions(3 cases, roots of auxiliary equation)
non-homog. eqns: trial and error method, resonance.

9/18- general 2nd order linear eqns- existence/uniqueness theorem
reduction of  order; example
non-homog. eqns- variation of parameter, example (Euler eqns)
discussion of HW 2 solutions (start)

9/23 - discussion of HW 2 solutions (end)
boundary-value problems
superposition principle
Cauchy-Euler equations

9/25 - Laplace transforms: definition, properties, solution of IVPs

9/30-Laplace transforms: examples
IVPs with discontinuous forcing term (without Laplace)
discussion of HW3

10/2- EXAM 1 (covers classes from  8/21 to 9/23)
Exam1-problems (PDF)

10/7 Laplace transforms: discontinuous functions, use of step functions
periodic functions
discussion of exam 1

10/9  Laplace transforms: periodic functions
convolution theorem

10/14 Linear homogeneous systems
discussion of exam 1

10/16  FALL BREAK (no class)

10/21    EXAM 2 (same material as exam 1)

10/23    no class

10/28    homog. systems- sinks/sources, complex ev's

10/30    homog. systems- 0 ev ,  purely imaginary ev's (conserved quantity)
fundamental matrix
non-homg. systems- variation of parameters formula

11/4       discussion of HW4 and HW5

11/6         EXAM 3 (Laplace transforms, systems) solutions
problems

11/11       Eigenvalues of second derivative on an interval
solution of heat eqn and wave  eqn by sepn of variables
(section 10.2)

11/13       Sections 10.3, 10.4:  Fourier series

11/18       Sections 10.4, 10.5: convergence, even/odd extensions,
heat equation, non-homog. problems

11/20       Section 10.6: wave eqn- formal solutions
(problems from text)

11/25   EXAM 4 (sections 10.2 to 10.6, as covered in class) problems-page 1
problems-page 2
solutions

11/27 THANKSGIVING (no classes)

12/2            Review (llast day)

12.9 FINAL (comprehensive)