Verify that y₁ = x and y₂ = x² are linearly independent solutions on the entire real line of the equation

x²y'' - 2xy' + 2y = 0,

but that W(x , x²) vanishes at x = 0. Why do these observations not contradict part (b) of Theorem 3?

THEOREM 3 Wronskians of Solutions Suppose that y, y2, ..., yn are n solutions of the homogeneous nth-order linear equation y() + P(x) y(n-1)+...+ Pn-1(x)y' + Pn(x) y = 0 on an open interval I, where each p, is continuous. Let (a) If y, 2, (b) If y. Y2.. **** (3) W = W(y, y2...., yn). yn are linearly dependent, then W = 0 on I. yn are linearly independent, then W #0 at each point of I. Thus there are just two possibilities: Either W = 0 everywhere on I, or W #0 everywhere on I.