Uniqueness of Solutions: Cauchy-Lipschitz Theorem

This lecture discusses the uniqueness of solutions in the context of differential equations, focusing on the Cauchy-Lipschitz theorem. The instructor begins by defining the concept of uniqueness for maximal solutions, explaining that a maximal solution is unique if all local solutions coincide on their defined intervals. The lecture then introduces the Cauchy-Lipschitz theorem, detailing the conditions under which a unique local solution exists. The instructor emphasizes the importance of the Lipschitz condition, which ensures that the difference between solutions can be controlled. Several examples illustrate the application of the theorem, demonstrating how local and global Lipschitz conditions affect the existence and uniqueness of solutions. The discussion includes cases where functions are not globally Lipschitz and how local conditions can still yield unique solutions. The lecture concludes with a summary of the key points regarding the conditions necessary for ensuring the uniqueness of solutions in differential equations, highlighting the significance of the Lipschitz condition in this context.