Wronskian - Definition

Definition

The Wronskian of two functions f and g is W(f,g) = fg′–gf ′.

More generally, for n real- or complex-valued functions f1, ..., fn, which are n − 1 times differentiable on an interval I, the Wronskian W(f1, ..., fn) as a function on I is defined by

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},\qquad x\in I.

That is, it is the determinant of the matrix constructed by placing the functions in the first row, the first derivative of each function in the second row, and so on through the (n - 1)st derivative, thus forming a square matrix sometimes called a fundamental matrix.

When the functions fi are solutions of a linear differential equation, the Wronskian can be found explicitly using Abel's identity, even if the functions fi are not known explicitly.

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