MATH 431-DIFFERENTIAL EQUATIONS II-FALL 2014- COURSE LOG
Th 8/21 General 1st-order equations: existence-uniqueness,
(KP 1.1, 1.2)
Due 8/28: (KP 1.6) 1.2, 1.8/1.16, 1.12, 1.13(i), 1.18
See also: 1.3, 1.4, 1.9, 1.19, 1.23
Honors: 1.14, 1.15
Tu 8/26 Autonomous equations: geometric analysis (KP 1.3)/ Autonomous
systems, phase-plane diagrams (KP 3.1, 3.2)
analysis of autonomous equations
MATLAB m-files for ODE: dfield and pplane
If you have Matlab installed on your computer, you can use this
software without knowing anything about Matlab.
Download it from this URL:
As an encouragement: students
turning in with their HW a relevant MATLAB plot for one
of the problems (with a caption describing what it shows) will get an
extra 1/2 point per HW set.
Th 8/28 Autonomous systems: existence/uniqueness; bounded solutions are
globally defined. Equilibrium points.
Phase-plane diagrams: nullclines. Examples: predator-prey, a
second-order equation. Conserved quantities.
Tu 9/2 Hamiltonian systems: conserved quantities. Ex: cubic 2nd order,
pendulum. Phase plane pictures.
Bounded invariant regions. Potentials and hamiltonians, relation with
Mechanics. Level sets of proper functions.
Ref: KP 3.1, W2.9
HW2 (due 9/9): KPsect 3.9: 3.2 (i),(ii), 3.8(i) (ii), 3.13 (competing
species), 3.15 (chemostat), 3.37 (Hamiltonian)
Hon: (KP sec 3.9): 3.7 (dipole), 3.14 (epidemic model)
Th 9/4 Predator-prey revisited. Phase plane pictures and stability for
linear systems.(KP 3.4, W 2.3, 2.4)
Tu 9/9 Handout:
Diagonalizable linear systems and stability.
Linear systems in 2D: eigenspaces, reduction to standard form,
Ref: [KP 3.3, W 2.3, 2.4]
Th 9/11 Linear systems in 2D: complex eigenvalues, real standard form.
for diagonalizable systems in n dimensions. Case of zero real
part. Example: coupled harmonic oscillators.
Ref: handout, [W 1.12]
HW3 [KP 3.9]: 9,10, 11, 12 (only item (i) in each); Waltman 1.12: 1.
Hon: the exercises proposed in the handout; Waltman 1.12: 2
(due Th 9/18)
Tu 9/16 Qualitative methods for NL autonomous systems: flow of a vector
field (semigroup property), stability of equilibria,
omega- and alpha-limit sets (examples). Liapunov functions for
equilibria. Ref: [KP 3.4, w 2.6]
Th 9/18 Liapunov functions (cont.) Liapunov's theorem on
stability/asymptotic stability, sublevel sets are invariant, LaSalle
Examples (inc. pendulum with friction.)
Stability of equilibria and Liapunov functions
(7 pages, final version posted 9/30)
HW4 (due Tu 9/30) [KP sect 3.9]: 16, 17(i), 21, 23, 28(i)
Hon: [W, 2.6] 3ab, 5
Tu 9/23: class canceled
Th 9/25: linearization at an equilibrium (ref: [KP 3.5, W 2.5])
Tu 9/30: Examples: sum of periodic functions (from HW), gradient
systems, Predator-prey systems (start) [ref: [W, 2.8]).
HW5 (due 10/7): KP sect 3.9: 25, 35, 36, 39, 44
of periodic functions (HW solution)
Th 10/2: Predator-prey systems (cont.), periodic solutions
(start): KP 3.6, W 2.7
examples of autonomous systems in the plane
(7 pages, posted 10/7)
Tu 10/7: Review/questions
Th 10/9: Discussion of examples in handout/ Poincare'-Bendixson theorem
Tu 10/14: First test
Th 10/16: fall break (no classes)
Tu 10/21: existence/nonexistence of periodic solutions:
Poincare-Bendixson, closed orbits must enclose an equilibrium,
index of a vector field along a closed curve and local index at a
singularity (amazing theorem)/Bendixson-Dulac criterion.
Th 10/23: equations of Lienard and van der Pol[KP, p.131]
on Lienard's equation
HW6 (due 10/30): 45, 47, 52, 53, 54 (i), 57
PLAN for the second part of the course.
1. 2nd order equations: existence-uniqueness for the IVP, the
Wronskian, self-adjointness (intro): [KP 4.1]
2. Oscillation and comparison theorems, disconjugacy and uniqueness [W
4.3, KP 5.5]
3. Non-homogeneous linear BVPs, Green's functions. [W 4.2, KP 5.9]
4. Sturm-Liouville eigenvalue problems: existence of eigenvalues,
properties, eigenfunction expansions
[W 4.4, 4.5, 4.6, 4.8; KP 5.4]
Tu 10/28: E/U for the IVP, Wronskian, self-adjointness (KP 5.1)
Problems: KP 5.1, 5.2, 5.3, 5.4, 5.8
HW7 (due 11.11): 5.1 (iii),
Th 10/30: disconjugacy, Green's functions, existence for non-hom BVP
Problems: KP 5.62, 5.65, 5.67, 5.68, 5.69
HW7; (due 11/11): 5.62(iii),
5.65, 5.68 (iii), (iv)
(version date: 11/4)
Tu 11/4: Green's functions: examples; Liapunov's criterion for
Take-home test collected.
Th 11/6: Liapunov criterion-examples; Oscillation/comparison
DISCUSSION PROBLEMS: from now
until the end of the course I will list a small number
of more challenging problems, with written solutions to be handed in on
the last day of classes.
You may work on those with a classmate, or ask questions about them in
class or during office hours.
The grade on this total assignment will enter the average with the same
weight as a test.
The weights will be: HW=20%, high two of three tests: 20% each,
Discussion Problems: 20%, Final:20%
(This is optional: if you decide not to turn it in, or if it would
lower your average, I will compute the average with
the originally announced weights instead.)
First list: KP 5.4, 5.69, W
Tu 11/11: Oscillatory properties of solutions
HW 8 (due 11/18): KP 5.16 (iii), (v); 5.17, 5.29; W, p.224 no. 4.
Th 11/13: Existence of SL eigenvalues/Properties of SL eigenfunctions/
Tu 11/18: Definitions/examples: metric space, normed vector space,
completeness, Banach space, Hilbert space.
Contractions in complete metric spaces have unique, globally attracting
Th 11/20: Picard's existence/uniqueness theorem; Lipschitz condition,
Picard iteration, error estimate.
Take-home test given.
Problems (from [W, Ch, 3.3]: 2,3,4,5/
Ch 3.4: 1(e)(g)
[KP, 8.11]: 9,10,11,12,13
Discussion problems: [W, Ch
HW9: problems in boldface listed above.
Tu 11/25: extensions of Picard's theorem: global existence in globally
Lipschitz or linear cases/ Nonlinear BVPs [W, 3.6]
Take-home test collected
Discussion problems: [W, 3.6]:
to two discussion problems
Tu 12/2 Discussion of problems/ examples. Gronwall's inequality
HW 9 due
Final Exam: Monday, 12/8: 10:15-12:15.
Suggestions: The entries in the course log give an idea of which
topics I think are important; for a summary of the theory,
read the online handouts. To review for the final: try to solve the
problems on the three tests and the homework, without looking at the
solutions. The final will be based on the problems found in the tests
and the homework sets, with small variations.