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 function values can be bounded by a constant. Several examples illustrate the application of the theorem, including cases where functions are globally Lipschitz and where local Lipschitz conditions lead to unique solutions. The discussion also highlights scenarios where uniqueness fails, particularly when dealing with functions that exhibit infinite slopes. The lecture concludes by reinforcing the significance of correctly defining the intervals and conditions to ensure the uniqueness of solutions in differential equations.