Existence and Uniqueness Theorem

This lecture covers the existence and uniqueness theorem of a maximal solution to a Cauchy problem, stating that there exists a unique maximal solution within a certain interval, satisfying specific continuity and Lipschitz conditions. The theorem guarantees the uniqueness and maximality of the solution, which is proven through the analysis of local solutions and the behavior of the maximal solution. The lecture also explores the concept of local solutions and the conditions under which a solution is considered maximal.