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:

$c_{1}{\vec {x}}^{(1)}+\ldots +c_{k}{\vec {x}}^{(k)}=0$

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

$W=Cexp[\int trP(t)dt]$

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