Ordinary Differential Equations: Definitions and Examples

This lecture introduces ordinary differential equations (ODEs) by defining them as equations relating a variable, an unknown function, and its derivatives. The concept of the order of an ODE is explained, along with examples illustrating different orders and solutions. The notion of a general solution and the distinction between singular and general solutions are discussed. The lecture also covers separable first-order ODEs, showing how to solve them by integrating. Various examples are provided to demonstrate the process of finding general solutions. The instructor emphasizes the importance of understanding the general solution and solving ODEs with initial conditions, known as Cauchy problems.