Differential Equations: Methods and Solutions

This lecture covers methods for solving first-order linear differential equations, focusing on separation of variables and the integrating factor method. The instructor begins by addressing previous class logistics and then delves into the concept of maximal solutions, emphasizing the importance of the solution's domain. The lecture explains the separation of variables method, illustrating its application to specific equations. The instructor introduces the integrating factor method, detailing how to find the general solution of a linear differential equation. Examples are provided, including the Cauchy problem, where the instructor demonstrates the steps to derive solutions using both methods. The lecture also discusses the significance of initial conditions and how they affect the solution. The instructor highlights the advantages of the integrating factor method over separation of variables, particularly in terms of efficiency and rigor. Throughout the lecture, the instructor encourages students to practice both methods to gain a deeper understanding of differential equations and their solutions.