146A 

Office hours: See syllabus. For syllabus please refer to eLearn page under modules.

Email: nchie005@ucr.edu

Please feel free to contact me for any reason.

Color coding of notes:

• Black - Theorems, lemmas, context, writing in general

• Grey - Reserved for proofs 

• Green - Definitions

• Red - Examples, exercises

• Blue - Remarks, clarification, justification

Class notes will be continuously updated and you must be logged into your UCR email: Notes completed

The class material is based off of notes provided by Prof. Zhenghe Zhang. 

Week 1 

• Day 1 (June 22)

  • 1. Introduced definition of differential equations and natural questions that arise: motion of fluids, quantum particles, solar systems, etc. 

    2. Introduced Theorem 7.1.1, which is a uniqueness and existence theorem. 

    3. Answered why continuity of f is necessary. In particular, we construct an example of a discontinuous f that leads to no solution to an IVP.

• Day 2 (June 23)

  • 1. Answered why continuity of partial_y(f) is necessary.

    2. Answered why h was necessary in Theorem 7.1.1.

    3. Provided an application of Theorem 7.1.1.

• Day 3 (June 24)

  • Discussion section - Notes (small typo: should state m-rows and n-columns)

    1. Review of math 46: 1st order linear (integrating factor method, bernoulli equations, and separable equations.

    2. Review of matrix: notation and operations,  determinants for 2x2 and 3x3 matrices.  inverses of matrices. and solving eigenvalues and eigenvectors.

    3. Review of matrix functions and properties like continuity and differentiability.

    4. Review of vector spaces and vector subspaces.

  • Lecture

    1. Discussed global existence and uniqueness theorem for system of linear ODE's.

    2. Proved that the solution set of system of linear ODE forms a vector space.

    3. Defined a basis and constructed a set of size n of linearly independent solutions.

• Day 4 (June 25)

  • 1. Completed the proof that the solution set for a vector space of dimension n.

    2. Discussed general solutions of systems of n-linear ODE's.

    3. Discussed fundamental matrix and how one can equivalently write a general solution.

    4. Discussed how the wronskian implies linearly independent.

    5. Proved key property that wronskian is always 0 or always nonzero.

Week 2

• Day 5 (June 29)

  • 1. Discussed theorems relating eigenvalues and eigenvectors to solutions to constant coefficient matrix ODE.

    2. Described the three cases that arise for the constant coefficient 2x2 matrix case.

    3. Discussed notation to describe the three cases in completion.

    4. Began first case and went over one example.

• Day 6 (June 30)

  • 1. Discussed another example problem for real-valued and distinct roots.

    2. Discussed 3x3 case with 2 real-valued roots where one has multiplicity 2 and dimension of the eigenspace is 2.

    3. Discussed the complex roots case. Proved how to find two real-valued solutions from the complex solution admitted from Theorem 7.5.1.

• Day 7 (July 1)

  • Discussion section 

    1. QUIZ 1 - 3 questions and 30 minutes based on HW1 material (regrades close Friday at 5pm)

    2. Review of class so far.

    3. Quiz 1, which was based off of question 2,3, and 4 from homework 1.

  • Lecture

    1. Briefly reviewed complex root case.

    2. Discussed repeated root case with dimension of eigenspace being 1.

    3. Proved the generalized eigenvector is linearly independent from original eigenvector.

    4. Proved a defined solution set is linearly independent, which provides the fundamental matrix.

• Day 8 (July 2) (Zoom notes)

  • 1. Review of section 7.5,7.6, and 7.8.

    2. Went over 3x3 example.

    3. Introduced equilibirum solutions and points.

    4. Discussed characterizing equilibirum points briefly.

    5. Defined parametric curves.

Week 3

• Day 9 (July 6)

  • 1. Briefly reviewed the introduction of section 9.1 and 9.2.

    2. Sketched phase portraits for case 1 of eigenvalue, which was both negative.

    3. Began case 2 of eigenvalues, which was the both positive case.

• Day 10 (July 7)

  • 1. Sketched phase portrait for case of eigenvalues both positive.

    2. Sketched phase portrait for case of eigenvalues of different signs.

    3. Sketched phase portraits for complex eigenvalues with different conditions on the real part of the root.

    4. Began some examples problems.

• Day 11 (July 8)

  • Discussion section 

    1.  QUIZ 2 - 2 questions and 30 minutes based on HW2 and half of HW3 material

    2. Worksheet problems consisting of finding general solutions, solving IVPs, a general version of HW2 Q2, and sketching phase portraits.

  • Lecture

    1. Concluded the example problems of sketching phase portraits.

    2. Discussed degenerate cases.

    3. Classified type of point depending on discriminant and trace.

    4. Began discussing techniques to handle nonlinear systems, linearization  near equilibria.

• Day 12 (July 9) 

  • 1. MIDTERM - Based on all material up to and including, section 7.8. Will not include section 9 onwards.

Week 4

• Day 13 (July 13)

  • 1. Discussed linearization and went over examples.

    2. Began undamped pendulum example and will continue in the following lecture.

• Day 14 (July 14)

  • 1. Completed undamped pendulum.

    2. Discussed first global technique, which is utilizing polar coordinates.

    3. Briefly introduced the second technique, which is using exact equations.

• Day 15 (July 15)

  • Discussion section - 

    1. QUIZ 3 - Based on HW3 and HW4, 2 questions and 30 minutes.

    2. Midterm solutions.

  • Lecture

    1. Completed global technique examples using exact equations.

• Day 16 (July 16) 

  • 1. Began competing species model.

Week 5

• Day 17 (July 20)

  • 1. Continued competing species model.

    2. Sketched phase portrait for 3 out of 4 cases depending on the positions of the x and y nullclines.

• Day 18 (July 21)

  • 1. Completed final case for competing species model.

    2. Setup the model for predator-prey and began the work to sketch the phase portrait.

• Day 19 (July 22)

  • Discussion section - 

    1. QUIZ 4 - Based on HW4 and HW5

    2. Work on answering conceptual questions to check understanding of the course.

  • Lecture

    1. Complete phase portait sketch of predator-prey model.

    2. Any additional time is dedicated to any questions.

• Day 20 (July 23) 

  • 1. Review/summary of entirety of class.

• Finals day (July 25, see syllabus for time and location)