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)