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This tag is for various questions relating to "Wronskian". In mathematics, it is a determinant used in the study of differential equations, where it can sometimes show linear independence in a set of solutions.

Definition: The Wronskian(or Wrońskian) of $$~n~$$functions $$y_1(x),~y_2(x),\cdots,y_n(x)~$$is denoted by $$~W(x)~$$or,$$~W(y_1,~y_2,\cdots,y_n)(x)~$$and is defined to be a determinant $$W(x)~=~W(y_1,~y_2,\cdots,y_n)(x)~=~\begin{vmatrix} y_1 & y_2 & \cdots & y_n \\ y'_1 & y'_2 & \cdots & y'_n \\ .\\ .\\ .\\ y^{(n-1)}_1 & y^{(n-1)}_2 & \cdots & y^{(n-1)}_n \end{vmatrix}$$

Applications:

Let $$y_1(x)$$ and $$y_2(x)$$ be two real valued differentiable functions on a set $$S = [a,b]$$ (say). If Wronskian $$W(y_1,y_2)(t_{0})$$ is nonzero for some $$t_{0}$$ in $$[a,b]$$, then $$y_1$$ and $$y_2$$ are linearly independent on $$[a,b]$$.

If $$y_1$$ and $$y_2$$ are linearly dependent then the Wronskian $$W(y_1,y_2)(t_{0})$$ is zero for all $$t_{0}$$ in [a,b] .

Generalization:

Let $$y_{1}$$ and $$y_{2}$$ be functions of two independence variables $$x_{1}$$ and $$x_{2}$$ i.e., $$y_{1} = y_{1}(x_{1} ,x_{2})$$ and $$y_{2} = y_{1}(x_{1} ,x_{2})$$ for which all partial derivatives of $$1^{st}$$ order, $$\frac{\partial y_{1}}{\partial x_{k}}$$, $$\frac{\partial y_{2}}{\partial x_{k}}$$, $$(k = 1,2)$$ exists throughout the region $$A$$. Suppose, farther, that one of the functions, say $$y_{1}$$, vanishes at no point of $$A$$. Then if all the two rowed determinants in the matrix $$\begin{pmatrix} y_{1} & y_{2} \\ \frac{\partial y_{1}}{\partial x_{1}} & \frac{\partial y_{2}}{\partial x_{1}} \\ \frac{\partial y_{1}}{\partial x_{2}} & \frac{\partial y_{2}}{\partial x_{2}} \end{pmatrix}$$ vanish identically in $$A$$, $$y_{1}$$ and $$y_{2}$$ are linearly dependent in $$A$$, and in fact $$y_{2}=c y_{1}$$.

References:

https://en.wikipedia.org/wiki/Wronskian

http://mathworld.wolfram.com/Wronskian.html

"Green, G. M., Trans. Amer. Math. Soc., New York, 17, 1916,(483-516)".