Linear differential equations

This lecture covers linear differential equations of order n, defining them as equations of the form y^(n)(x) + g_n-1(x) y^(n-1)(x) + ... + g_1(x) y'(x) + g_0(x) y(x) = f(x). It explores homogeneous and inhomogeneous equations, solutions, linear independence, and the existence and uniqueness theorem. The lecture also presents examples showcasing linearly independent functions like cos(x) and sin(x). Theorems are introduced to explain the existence of n linearly independent solutions for an nth-order homogeneous equation, and the general solution is derived from n linearly independent solutions. The proofs are briefly outlined.