Linear Differential Equations: Solutions and Wronskian

This lecture focuses on linear differential equations of the second order, particularly those with constant coefficients. The instructor begins by reviewing the general solution for homogeneous equations and introduces the concept of the Wronskian, which is a determinant used to determine the linear independence of solutions. The lecture covers the characterization of two solutions of a linear homogeneous differential equation, emphasizing that if the Wronskian is non-zero, the solutions are linearly independent. The instructor demonstrates how to compute the Wronskian for specific solutions and discusses the implications of linear dependence and independence. The lecture also addresses the method of variation of parameters for finding particular solutions to non-homogeneous equations. Throughout the session, the instructor provides examples and exercises to illustrate the concepts, ensuring that students understand how to apply these methods to solve differential equations effectively. The importance of these techniques in mathematical analysis and their applications in various fields is highlighted.