Bob is again thinking about the theorem from class about sol

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.

Solution

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.