Syllabus & Concept of Teaching Talk:
Course:
Ordinary differential equations (4th semester)
Purpose/Objective:
Introduce the central theoretical result on existence and uniqueness of ODEs.
The essence of its proof, the sharpness and generality of the theorem.
Content:
1) Introducing the concept of existence and uniqueness of solutions: Recapitulation of the two key
examples: u = u2, u(0) = 1 (limit of existence interval), u =√ , u(0) = 0 (non-uniqueness).
2) Introduction of the theorem. Mention that we proof it later. Discussion of its relation to the examples.
Discussion of the case of functions that are globally Lipschitz in relation to linear ODE.
3) Discussion on the proof strategy–using fixed points. The Banach fixed point theorem (withproof)
4) Proof of the Banach fixed point theorem for the case of globally Lipschitz function.
5) Proof of the general case.