### Math 322

##### Ordinary Differential Equations
Spring 2022

Current week   |   Moodle   |   Course information |   Presentations
Instructor: David Perkinson (schedule)
Course lecture notes and homeword: 322 Lecture notes, Homework (last update (4/27/22).
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Text: Differential equations and dynamical systems, by Lawrence Perko (TOC).
Week 1: January 24 - 28
Monday: Separable equations. (lecture, tex file, slides).

Wednesday: Logistic equation. Homogeneity trick for separable equations. Exact equations (if time). (lecture, tex file, slides).

Friday: Exact equations. Integrating factors. (lecture, tex file, slides).
Week 2: January 31 - February 4
Monday: First-order linear. Linear homogeneous constant coefficients. (lecture, tex file, slides).

Wednesday: Bernoulli equation motivation. Linear homogeneous constant coefficients: complex roots and repeated roots. Method of undetermined coefficients. (lecture, tex file, slides).

Friday: Special second-order equations. (lecture, tex file, slides).
Week 3: February 7 - 11
Monday: Matrix exponentiation. (lecture, tex file, slides).
• Reading: Sections 1.1–1.3. (If you are not using edition 3, make sure you are reading the correct sections by consulting the TOC—there is a link at the top of this webpage.)
• Turn in: HW2 (Overleaf template)

Wednesday: Fundamental theorem for linear systems. (lecture, tex file, slides).

Friday: Fundamental theorem for linear systems. Linear systems in $$\mathbb{R}^2$$. (lecture, tex file, slides).
Week 4: February 14 - 18
Monday: Linear systems in $$\mathbb{R}^2$$. (lecture, tex file, slides).

Wednesday: Jordan form. (lecture, tex file, slides).

Friday: Exponentiating Jordan matrices. Algorithm for computing Jordan form. (lecture, tex file, slides).
Week 5: February 21 - 25
Monday: Stability theory. Linear systems in $$\mathbb{R}^3$$. Nonhomogeneous equations. (lecture, tex file, slides).

Wednesday: Nonhomogeneous equations. (lecture, tex file, slides).
• Reading: Section 1.10 (and the lecture notes).

Friday: Higher-order homogeneous linear equations with constant coefficients. (lecture, tex file).
Week 6: February 28 - March 5
Monday: Higher-order homogeneous linear equations with constant coefficients. (lecture, tex file, slides).

Wednesday: Existence and uniqueness for non-linear systems. (lecture, tex file, slides).
• Reading: Sections 2.1 and 2.2.

Friday: Existence and uniqueness for non-linear systems. (lecture, tex file, slides).
• Reading: Sections 2.1 and 2.2.
Week 7: March 8 - 11
Monday: Existence and uniqueness for non-linear systems. (lecture, tex file, slides).

Wednesday: Linearization. (lecture, tex file, solutions, solutions tex file, awesomeness).
• Reading: Sections 2.1 and 2.2.

Friday: Dependence on parameters, maximal interval. Begin stable manifold theorem. (lecture, tex file, slides).
Week 8: March 14 - 18
Monday: Stable manifold theorem. (lecture, tex file, slides).

Wednesday: Stable manifold theorem. (lecture, tex file, solutions).

Friday: Hartman-Grobman theorem. (lecture, tex file, slides).
• Reading: Sections 2.7 and 2.8.
Week 9: March 28 - April 2
Monday: Stability and Liapunov functions. (lecture, tex file, slides).

Wednesday: No class today

Friday: Liapunov functions. (lecture, tex file, slides).
Week 10: April 4 - 8
Monday: Planar systems. (lecture, tex file, slides).
• Reading: Sections 2.10 and 2.11.
• Turn in: HW9 (Overleaf template)
• Example due to Peron (example 5 in Perko reading). A node turns into a focus upon the addition of nonlinear terms: node, focus.

Wednesday: Global theory for nonlinear systems: index theory. (lecture, tex file, slides).

Friday: Global theory for nonlinear systems: index theory. (lecture, tex file).
Week 11: April 11 - 15
Monday: Global theory for nonlinear systems: index theory. (lecture, tex file, slides).

Wednesday: Critical points at infinity, and global phase portraits. (lecture, tex file, slides).

Friday: Critical points at infinity, and global phase portraits. (lecture, tex file, slides).
Week 12: April 18 - 22
Monday: Critical points at infinity, and global phase portraits. (lecture, tex file, slides).

Wednesday: Resolution of singularities. (lecture, tex file, no slides).
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