Separation of Variables: Solving Differential Equations

This lecture introduces the method of separation of variables for solving differential equations. The instructor begins by establishing the theory related to the existence and uniqueness of solutions. The focus is on defining the types of functions involved in the problem, particularly continuous functions. The lecture explains how to separate variables in equations, allowing for the integration of each variable independently. The instructor emphasizes the importance of ensuring that certain functions do not vanish, which is crucial for the existence of a unique local solution. The construction of the solution is detailed, showing how to derive it from the defined functions. The lecture also discusses the continuity of these functions and their inverses, which is essential for establishing the uniqueness of the solution. Finally, the instructor verifies that the constructed function satisfies the original differential equation and the initial conditions, concluding with a preview of upcoming examples that will apply this method in practice.