Scalar Linear Higher Order ODEs

This lecture covers scalar linear higher order ordinary differential equations with possibly x-dependent coefficients, focusing on properties of solutions, existence, uniqueness, and the equivalence with systems of first-order ODEs. It discusses the bijection between solutions of scalar ODEs and systems of ODEs, the superposition principle, and the relationship between solutions to homogeneous and inhomogeneous systems. The lecture emphasizes the existence and uniqueness of solutions, the formation of a vector space of solutions, and the conditions for solutions to be linearly independent. It also explores the concept of a general solution to inhomogeneous problems.