Question

Suppose you solved a second-order equation by rewriting it as a system and found two scalar solutions: y = ' and = 24. Think of the corresponding vector solutions y, and y, and use the Wronskian to show that the solutions are linearly independent. Wronskian - det These solutions are linearly independent because the Wronskian is Choose for all X.

Answer 1

To show that the scalar solutions y = ' and = 24 are linearly independent, we can use the Wronskian.

Step-by-step explanation:

To show that the scalar solutions y = ' and = 24 are linearly independent, we can use the Wronskian. The Wronskian is the determinant of a matrix formed by the two vector solutions y₁ and y₂. If the Wronskian is non-zero, the solutions are linearly independent.

Let's denote the vector solutions as y₁ and y₂. In this case, y₁ = [', 1] and y₂ = [24, 0].

To calculate the Wronskian, we form a matrix with the components of y₁ and y₂: [[', 24], [1, 0]]. Taking the determinant, we get: det([[', 24], [1, 0]]) = ('*0) - (1*24) = -24.

Since the Wronskian is non-zero (-24 ≠ 0), we can conclude that the solutions y = ' and = 24 are linearly independent.