Math 123: Ordinary Differential Equations

Instructor: Di Fang         Please read the Syllabus carefully!

Lecture: Tuesdays and Thursdays, 2:00--3:29pm, 3111 Etcheverry ⇒ Berkeley time!

Office Hours: Tue 3:40-4:39pm and Wed 3:00-3:59pm, 843 Evans Hall ⇒ Not Berkeley time!

Textbook: The Qualitative Theory of Ordinary Differential Equations by Fred Brauer and John A. Nohel.

Email: ⇒ Please kindly include Math 123 in the subject line when emailing.


Course Outline:

  • Teminology; Review: techniques and tricks; the need for theory
  • Theory of ODEs: existence, uniqueness, continuation.
  • Linear Systems (with an intro to phase space analysis)
  • Stability theory and Lyapunov method
  • Applications and additional topics as time permits, e.g. biological models (flocking of birds), control theory.
  • (Things are subject to change. Once it happens I will let you know.)


Week Tuesday Topics Thursday Topics HW (posted on bCourse)
    1 Policy, Philosophy, Terminology
(not all in book) Lecture Notes
Happy labor day!
    2 How to solve ODEs? A Review of some techniques from [Calc/54] -- Old and New Lecture Notes Review of techniques continued, some basic theorem/proof of the structure of the solution space, the need for theory [BN 1.5]
Lecture Notes
HW1 (due Sept 10th)
    3 (Picard's) Existence theorem & proof (under the Lipschitz condition) [BN 3.1] -- Picard iteration, Uniform Convergence Finish proof & (Peano's) Existence theorem, generalization to higher Dimension [BN 3.2], Gronwall Ineq [BN 1.7] & Uniqueness Theorem [BN 3.3]
HW2 (due Sept 19th)
    4 Continuation of Solutions, Global existence, a priori estimate [BN 3.4] Dependence on initial condition and parameters [BN 3.5], Baby sensitivity analysis of Uncertainty Quantification, How equilibrium depends on parameter -- an intro to bifurcation analysis
Lecture Notes
HW3 (due Sept 26th)
Chaotic system (sensitivity of initial data) -> try it here!
    5 Existence theorem of system of linear ODEs [BN 2.2], vector norm, matrix norm [BN 1.4/2.1], a review of vector space
Lecture Notes
Solution space of linear homogeneous systems, fundamental matrix, Abel's formula [BN 2.3] (on textbook, lecture notes will not be posted) HW4 - fixed typo (due Oct 3rd)
    6 Non-homogeneous system, Variation of constant formula [BN 2.4] Linear system with constant coefficients, matrix exponenetial [BN 2.5] HW5 (due Oct 10th)
    7 More on complex eigen-pairs (using geometric meaning of rotation matrices); More on Jordan Block
partial Lecture Notes
No Class due to Power Outrage No HW! Midterm is coming!!
Midterm is coming! Good luck everyone!
    8 Midterm Fundamental matrix with general case - Jordan Canonical form, generalized eigenvectors, Jordan chains [BN 2.6] & on Linear Algebra! No HW!
    9 Asymptotic Behavior of Solutions of Linear System with Constant Coefficients -- estimate for homogeneuous & non-homogeneous cases [BN 2.7] Lecture Notes Q&A on Midterm HW6 (due Oct 31st) Happy Halloween!
    10 A taste of coercivity, 2D linear autonoumous system (an intro to Phase space analysis) [BN 2.8] Linear system with periodic coefficients, Floquent Theorem, Floquet multiplier [BN 2.9] HW7 (due Nov 7th) Happy Halloween!
    11 Finish linear system with periodic coefficients (non-homo case), periodic solutions [BN 2.9], intro to Lyapunov stability [BN 4.1] stability theorem for linear constant coefficient system, variable coefficients -- perturbation method [BN 4.2-4.3] HW8 (due Nov 14th)
    12 Stability of almost linear system - linearization + perturbation [BN 4.4], bootstrapping argument Lecture Notes Topological conjugate, Hartman-Grobman theorem (Ref: L. Perko, Differential equations and dynamical systems Chapter 2.8) HW9 (due Nov 21th)
    13 Lyapunov direct method, theorem proof [BN 5.2, 5.3] Lyapunov method continued, LaSalle Invariance Principle, Hamiltonian system HW10 (due Dec 3rd) Last homework of this semester!
    14 Biological applications: Flocking of the birds (Reference Papers: [Tadmor-Ha 2008], [Liu-Ha 2009]) HAPPY THANKSGIVING!
    15 Flocking Estimate continued (quantatitive rate via Dissipative inequality and Lyapunov) Review and Q&A (let me know of the topics that are most confusing, and would like to go over again)
Final! Good luck everyone!



True or False: I do not need to take notes in class, because Di will always post her notes.

False!! Di is posting the notes for the first three lectures for easier reference because it was a review and material is not in the book. So take notes after the first three lectures.

(By Di on Sept. 3)

Wecome to the class. Wish we have a pleasant and productive semester together!

Please email me about the time conflicts with the office hour by Monday (Sept. 1st), so that we could settle the final office hours.

(By Di on August. 30)