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 related to the Cauchy problem, emphasizing the importance of initial conditions. The lecture explains how to partition intervals and the significance of local solutions, which are defined on subintervals. The concept of maximal solutions is introduced, highlighting the conditions under which solutions can be extended. The instructor presents the Cauchy-Peano theorem, which states that if a function is continuous, there exists at least one local solution. The lecture also differentiates between local and global solutions, explaining the implications of each. Examples are provided to illustrate the behavior of solutions under different initial conditions, emphasizing the existence of solutions in various scenarios. The lecture concludes with a discussion on the uniqueness of solutions, setting the stage for further exploration in subsequent lectures.