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Concept# Wronskian

Summary

In the mathematics of a square matrix, the Wronskian (or Wrońskian) is a determinant introduced by the Polish mathematician . It is used in the study of differential equations, where it can sometimes show linear independence of a set of solutions.
Definition
The Wronskian of two differentiable functions f and g is
W(f,g)=f g' - g f'
.
More generally, for n real- or complex-valued functions f1, …, fn, which are n – 1 times differentiable on an interval I, the Wronskian
W(f_1,\ldots,f_n)
is a function on
x\in I
defined by
W(f_1, \ldots, f_n) (x)=
\det
\begin{bmatrix}
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{bmatrix}.
This is the determinant of the matrix constructed by placing the functions in the first row, the first derivatives of the functions in the second

Official source

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