Bob is again thinking about the theorem from class about \"solutions to a homoge
ID: 3404051 • Letter: B
Question
Bob is again thinking about the theorem from class about "solutions to a homogeneous linear differential equation being linearly independent if and only if their Wronskian is always nonzero". This time he notes that x and x^2 are solutions to the same homogeneous linear differential equation y'" + x^2y" - 2xy' + 2y = 0, and are linearly independent, but their Wronskian is W(x, x^2) = x^2, which is not always nonzero. Bob again feels like he has found a counterexample to the theorem. Explain why Bob is again wrong.Explanation / Answer
If two functions f(x) and g(x) are differentiable on some interval I. then the functions are linearly independent if their Wronskian is not zero for some values of x on I.
Using this concept, the functions x and x^2 are differential on (-inf, inf), and the Wronskian W(x, x^2)=x^2 is zero only if x=0. That means, the wronskian is not zero for all values of x in (-inf, inf) and hence, the solutions x and x^2 are linearly independent.