Linear Independence: The Wronskian Concept

This lecture discusses the concept of the Wronskian associated with two solutions of a differential equation. The instructor explains that two solutions are linearly independent if every linear combination of these solutions equals zero only when the coefficients are zero. The Wronskian, denoted as W, is defined as a function that involves the first solution multiplied by the derivative of the second solution, minus the derivative of the first solution multiplied by the second solution. This function can be interpreted as the 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 the proof of this theorem through a series of logical steps, including a proof by contradiction, establishing the equivalence between the linear independence of solutions and the non-vanishing of the Wronskian.