# Ordinary Differential Equations, Fall 2022

 Lecturer: Grigorios Fournodavlos Office: E.314 E-mail: gfournodavlos AT uoc DOT gr

### Course info

Curriculum, prerequisites, and recommended books can be found here.

### Schedule and office hours

A.214: Monday/Friday 13:15-15:00.

E.314: Office hours Friday 15:00-17:00 after class during the semester.

### Catch-up classes (missed lessons)

E.204: Wednesday October 26, 11:00-13:00.

E.204: Wednesday November 2, 11:00-13:00.

### Notes

Hand-written notes from the lectures by Antonis Spanoudakis.

### Exercises

Sheet 1.gr, Sheet 1.en (Week 1)

Sheet 2.gr, Sheet 2.en (Week 4)

Sheet 3.gr, Sheet 3.en (Week 7)

Sheet 4

### Material covered

Week 1 (3/10/2022--7/10/2022)

First order ODEs, local existence and uniqueness, Picard iteration, continuous dependence on initial data and other parameters, Gronwall lemma, continuation of solutions.

See also Sections 2.1, 2.2.1, 2.3, 2.4 in the book by Alikakos & Kalogeropoulos.

Week 2 (10/10/2022--14/10/2022)

Solutions to some exercises in Sheet 1. First order systems (nonlinear), existence and uniqueness, reduction of nth order equations to first order systems. Homogeneous linear equations of nth order with constant coefficients, general solution, initial value problem, existence and uniqueness.

See also Sections 4.1, 4.2, 4.3, 4.6.1, 4.6.2 in the book by Alikakos & Kalogeropoulos.

Week 3 (17/10/2022--21/10/2022)

Inhomogeneous linear equations of nth order with constant coefficients, undetermined coefficients method. Linear first order systems, existence and uniqueness. Dimension of solution space, Wronskian.

See also Sections 4.7, 4.8, 6.1, 6.2 in the book by Alikakos & Kalogeropoulos.

Week 4 (24/10/2022--28/10/2022)

Fundamental matrix, exponential matrix. Homogeneous first order systems with constant coefficients and diagonalizable matrix. Solutions to some exercises in Sheet 2.

See also Sections 6.3.2, 6.6 in the book by Alikakos & Kalogeropoulos.

Week 5 (31/10/2022--4/11/2022)

Homogeneous systems with non-diagonalizable matrices, generalized eigenvectors. Inhomogeneous first order systems. Revision exercises for the last 2 weeks.

See also Sections 6.7, 6.9 in the book by Alikakos & Kalogeropoulos.

Week 6 (14/10/2022--18/11/2022)

Qualitative theory (one dimension): equilibrium points, phase diagrams, stability/instability, linearization, dynamical systems.

See also Sections 3.1, 3.2, 3.3, 3.4 in the book by Alikakos & Kalogeropoulos.

Week 7 (21/10/2022--25/11/2022)

Vector field flow (two dimensions). 1-parameter families of differential equations, bifurcation values and diagrams. Qualitative theory of first order systems (extending previous definitions). Stability/instability for linear systems via eigenvalues, existence and uniqueness for non-linear systems, orbits, phase space diagrams.

See also Sections 3.6, 10.1, 10.3, 10.4 in the book by Alikakos & Kalogeropoulos.