Ordinary Differential Equations: Separable Variables

This lecture covers ordinary differential equations (EDOs) of the form F(x, y, y(1),..., y(n)) = 0, where F is a continuous function defined on an open set in Rn+2. It explains the concept of separable variables in EDOs, where y: R→R is an n times continuously differentiable function. The lecture introduces the notion of a Cauchy problem, providing an EDO and an initial condition. It delves into the definition of an ordinary differential equation with separable variables (EDOv.s.), which involves equations of the form y' = f(x)g(y). Theorems are presented to establish conditions for the existence and uniqueness of solutions in separable variable EDOs.