Wronskian determinant

The Wronskian determinant is the determinant that consists of the solutions of an $n$-th order ODE, for $n$ sufficiently smooth functions:

$$W(f_1,f_2,\dots ,f_n)=\left|\begin{array}{cccc}{f}_1& f_2& \dots & f_n\\ f_1^{{}^{\prime }}& f_2^{{}^{\prime }}& \cdots & f_n^{{}^{\prime }}\\ ⋮& ⋮& \ddots & ⋮\\ f_1^{n-1}& f_n^{n-2}& \dots & f_n^{n-1}\end{array}\right|$$

Like regular determinants, if $W=0$, then it is linearly dependent. Otherwise the functions are linearly independent. Under the existence and uniqueness assumption, we have either $W\ne 0\forall x\in I$ or $W=0\forall x\in I$. What this means is that it largely suffices to check that the Wronskian is non-zero.