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, $W'+pW=0$

$W(t)=Ce^{\int pdt}$

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).

Systems

General solutions to systems of equations are expressed as the following:

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 c_1\vec{x}^{(1)}+\ldots+c_k\vec{x}^{(k)}=0 }

For them, the Wronskian is defined to be the determinant of X. We have another clause of Abel's theorem and the following

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=Cexp[\int trP(t)dt] }

We can then draw a similar conclusion as above: the Wronskian is either always 0 or never 0.