Determine whether these functions are linearly independent

Question

Give a pair of functions {v₁(x) = x sin x; v₂(x) = x|sin x|} on the interval I = [-π, π].

a. Determine whether these functions are linearly independent or linearly dependent.

b. Show that the Wronskian W(x) = W[v₁, v₂] exists and calculate its value on the interval (I).

c. Prove that {v₁(x) = sin x; v₂(x) = sin² x} cannot be a set of fundamental solutions of any second order linear differential equation of 2-nd order On (I).