Lecture
This lecture discusses the Cauchy problem and the existence of solutions to differential equations. It begins by reviewing previous concepts related to differential equations and their graphical representations. The instructor introduces essential vocabulary and theorems regarding the existence and uniqueness of solutions for the Cauchy problem. The lecture emphasizes the importance of defining intervals and conditions for solutions, including local and global solutions. The instructor explains how to construct solutions over specified intervals and the significance of extending these solutions. The Cauchy-Peano theorem is presented, which states that if a function is continuous, there exists at least one local solution. The lecture also covers the implications of having a non-global solution and the conditions under which solutions may explode or cease to exist. Examples are provided to illustrate these concepts, highlighting the differences between local and global solutions and the conditions that affect their existence and uniqueness.