Wronskian

The Wronskian of n equations is the determinant of the following matrix.

$W(y_{1},y_{2})(x)={\begin{vmatrix}f_{1}(x)&f_{2}(x)&\cdots &f_{n}(x)\\f_{1}'(x)&f_{2}'(x)&\cdots &f_{n}'(x)\\\vdots &\vdots &\ddots &\vdots \\f_{1}^{(n-1)}(x)&f_{2}^{(n-1)}(x)&\cdots &f_{n}^{(n-1)}(x)\end{vmatrix}}$

It can be used to check if solutions of differential equations are linearly independent (when the Wronskian is nonzero).

Properties

For the Wronskians of solutions of linear differential equations, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W'+pW=0 }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W(t)=Ce^{\int pdt} }

As such, the Wronskian is either always 0 (when C is 0 and the functions are linearly dependent) or never 0 (when the C is nonzero and the functions are linearly independent).