Cauchy Problem: Differential Equations and Initial Conditions

This lecture discusses the Cauchy problem in the context of differential equations. It begins by detailing the nature of differential equations, which express a relationship between an unknown function and its first derivative. The instructor explains that in practical applications, particularly in physics and engineering, additional conditions are often required to narrow down the infinite possible solutions to a differential equation. These conditions, known as initial conditions or Cauchy conditions, specify a value for the function at a particular point. The lecture provides examples to illustrate how these conditions can lead to unique solutions. The instructor emphasizes the importance of finding a constant that satisfies both the differential equation and the initial condition. Various cases are examined, including scenarios where the initial value is zero or negative, and how these affect the existence and uniqueness of solutions. The lecture concludes with a discussion on the implications of these conditions for the behavior of solutions over defined intervals.