# 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.

### Session for questions before the final exam

A.214: Tuesday January 10, 15:00-17:00.

### Final exam

Wednesday January 11, 17:00-19:00.

### Material for the exam

Everything from the notes and exercise sheets, apart from how to solve nth order equations. No need to read from the second half of page 13 to the first half of page 19 in the notes. Also, no need to solve exercises 3, 4, 5 in Sheet 2.

See also Sections 2.1, 2.2.1, 2.3, 2.4, 3.1, 3.2, 3.3, 3.4, 3.6, 6.1, 6.2, 6.3.2, 6.6, 6.7, 6.9, 10.1, 10.3, 10.4, 10.5, 10.6 in the book by Alikakos & Kalogeropoulos.

More precisely, make sure to study:
• First order equations: existence and uniqueness, continuation of solutions, maximal domain of definition, global solutions vs finite time blow-up, Gronwall's inequality, continuous dependence on initial data and other parameters.

• First order systems: reduction of nth order equations to first order systems. Linear first order systems: Wronskian and linear independence, fundamental matrix, constant matrices of simple and non-simple form, generalized eigenvectors, exponential matrix (solution formula for inhomogeneous systems).

• Qualitative theory of first order equations: equilibrium points, stability/instability, phase diagrams, bifurcation of 1-parameter families of equations and diagrams.

• Qualitative theory of first order systems: equilibrium points, orbits, phase space diagrams, dynamical systems (vector field flows), linearization, stability/instability via eigenvalues, Lyapunov stability.

### Notes

Hand-written notes from the lectures by Antonis Spanoudakis.

### Exam Sheet and Solutions

January: Exam.gr, Exam.en

June: Exam-2.gr

September: Exam-3.gr

### Exercise Sheets

Sheet 1.gr, Sheet 1.en (1st order equations)

Sheet 2.gr, Sheet 2.en (1st order systems and nth order equations)

Sheet 3.gr, Sheet 3.en (Qualitative theory of 1st order equations)

Sheet 4.gr, Sheet 4.en (Qualitative theory of 1st order systems)