Wronskian

The Wronskian is a determinant used in the theory of differential equations to analyze the linear independence of a set of functions. For n functions y1, y2, ..., yn defined on an interval I and differentiable up to order n−1, the Wronskian W(y1, ..., yn)(x) is defined as the determinant of the matrix whose i-th row consists of the derivatives of y_i up to order n−1: W(x) = det [ y_i^{(j−1)}(x) ] with i = 1,...,n and j = 1,...,n. For two functions, W(y1, y2) = y1 y2' − y1' y2.

A basic use of the Wronskian is to test linear independence: if W(y1, ..., yn)(x0) ≠ 0 at some point x0 in I, then the functions y1, ..., yn are linearly independent on I. The converse is not guaranteed for arbitrary functions, but it holds under additional regularity assumptions such as when the functions are solutions of a linear homogeneous differential equation with continuous coefficients.

For an nth order linear homogeneous differential equation y^{(n)} + p_{n−1}(x) y^{(n−1)} + ... + p_0(x) y = 0, Abel’s identity gives W'(x) = −p_{n−1}(x) W(x). Consequently, W(x) = C exp(−∫ p_{n−1}(x) dx). Thus either W is identically zero or never vanishes on I; moreover, a nonzero W at a point implies a fundamental set of solutions spanning the solution space.

The Wronskian extends to systems y' = A(x) y, where W satisfies W' = tr(A(x)) W. In this context, a nonvanishing W indicates a fundamental set of solutions.