In mathematics, the Wronskian of n {\displaystyle n} differentiable functions is the determinant of a matrix formed by the functions and their derivatives up to order n − 1 {\displaystyle n-1} . It was introduced in 1812 by the Polish mathematician Józef Wroński, and is used in the study of differential equations, where it can show the linear independence of certain sets of solutions.
Definition The Wronskian of two differentiable functions f {\displaystyle f} and g {\displaystyle g} is W ( f , g ) = f g ′ − g f ′ {\displaystyle W(f,g)=fg'-gf'} . More generally, for n {\displaystyle n} real- or complex-valued functions f 1 , … , f n {\displaystyle f_{1},\dots ,f_{n}} , which are n − 1 {\displaystyle n-1} times differentiable on an interval I {\displaystyle I} , the Wronskian W ( f 1 , … , f n ) {\displaystyle W(f_{1},\ldots ,f_{n})} is the function
W ( f 1 , … , f n ) ( x ) = | f 1 ( x ) f 2 ( x ) ⋯ f n ( x ) f 1 ′ ( x ) f 2 ′ ( x ) ⋯ f n ′ ( x ) ⋮ ⋮ ⋱ ⋮ f 1 ( n − 1 ) ( x ) f 2 ( n − 1 ) ( x ) ⋯ f n ( n − 1 ) ( x ) | {\displaystyle W(f_{1},\ldots ,f_{n})(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}}}
defined for all x ∈ I {\displaystyle x\in I} . This is the determinant of the matrix constructed by placing the functions in the first row, their first derivatives of the functions in the second row, and so on through the ( n − 1 ) {\displaystyle (n-1)} -st derivative, thus forming a square matrix. When the functions are solutions of a linear differential equation, the Wrońskian can be found explicitly using Abel's identity, even if the functions themselves are not known explicitly. (See below.)
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