Linear Differential Equations: Second Order Solutions

This lecture focuses on second-order linear differential equations, discussing their structure and solutions. The instructor introduces the general form of these equations and emphasizes the importance of initial conditions. The lecture explains the concept of linear independence among solutions, illustrating how two solutions can generate all possible solutions through linear combinations. The instructor also covers the existence and uniqueness of solutions, referencing the Cauchy-Lipschitz theorem. A significant portion of the lecture is dedicated to constructing solutions, particularly in cases where the second member is zero, leading to homogeneous equations. The instructor demonstrates methods for finding linearly independent solutions and provides a theorem for constructing new solutions from known ones. The lecture concludes with practical examples to reinforce the theoretical concepts discussed, ensuring a comprehensive understanding of second-order linear differential equations and their solutions.