Wronskian determinant

also just called `Wronskian

$$W(y_1,y_2,\dots ,y_n)(x)=\left|\begin{array}{cccc}{y}_1(x)& y_2(x)& \cdots & y_n(x)\\ y_1^{\prime }(x)& y_2^{\prime }(x)& \cdots & y_n^{\prime }(x)\\ ⋮& ⋮& \ddots & ⋮\\ y_1^{(n-1)}(x)& y_2^{(n-1)}(x)& \cdots & y_n^{(n-1)}(x)\end{array}\right|$$

for example if $n=2$

$$W(y_1,y_2)=\left|\begin{array}{cc}{y}_1& y_2\\ y_1^{\prime }& y_2^{\prime }\end{array}\right|$$

then $y_1,\dots y_n$ are linearly independent if and only if $W\ne 0$.
This is important because in solving a linear system of differential equations, we need a fundamental set of linearly independent solutions. This is analogous to solving a system of linear equations, where the system has a unique solution only if the coefficient matrix is invertible—that is, its determinant is nonzero.