Linear Independence: The Wronskian Concept

This lecture discusses the concept of the Wronskian, a theoretical notion that systematically associates two particular solutions of a differential equation. The instructor defines linear independence of two solutions, explaining that a linear combination of these solutions equals zero only if the coefficients are themselves zero. The Wronskian function, denoted as W, is introduced as a determinant of a 2x2 matrix formed by the solutions and their derivatives. The lecture presents a theorem stating that if two solutions are linearly independent, then the Wronskian is non-zero for all values in the interval where the equation is defined. The instructor demonstrates both directions of the theorem's proof, using a proof by contradiction to show that if the Wronskian is zero, the solutions must be dependent. The lecture concludes by establishing the equivalence between the linear independence of solutions and the non-vanishing of the Wronskian across the interval, solidifying the importance of this concept in differential equations.