According to Wikipedia, if the Wronskian of two functions is always zero, then they are not necessarily linearly dependent.
But it seems that if the two functions are solutions of the same homogeneous second-order linear differential equation, then the condition $W[y_1, y_2](t) = 0$ does indeed imply that they are linearly dependent.
Online, I found that if two functions are real analytic and their Wronskian is identically zero, then they are necessarily linearly dependent. But there is no reason that the solutions to a linear differential equation should be real analytic.
How can we prove that the condition $W[y_1, y_2](t) = 0$ implies the linear dependence of $y_1(t)$ and $y_2(t)$? More generally, how can we prove that the condition $W[y_1, \ldots, y_n](t) = 0$ implies the linear dependence of $y_1(t), \ldots, y_n(t)$?