Differential equations |
---|

Scope |

Classification |

Solution |

People |

In the mathematics of a square matrix, the **Wronskian** (or **Wrońskian**) is a determinant introduced by the Polish mathematician Józef Hoene-Wroński (1812). It is used in the study of differential equations, where it can sometimes show linear independence of a set of solutions.

The Wronskian of two differentiable functions *f* and *g* is .

More generally, for *n* real- or complex-valued functions *f*_{1}, …, *f _{n}*, which are

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 row, and so on through the derivative, thus forming a square matrix.

When the functions *f _{i}* are solutions of a linear differential equation, the Wronskian can be found explicitly using Abel's identity, even if the functions

If the functions *f _{i}* are linearly dependent, then so are the columns of the Wronskian (since differentiation is a linear operation), and the Wronskian vanishes. Thus, one may show that a set of differentiable functions is linearly independent on an interval by showing that their Wronskian does not vanish identically. It may, however, vanish at isolated points.

A common misconception is that *W* = 0 everywhere implies linear dependence, but Peano (1889) pointed out that the functions *x*^{2} and |*x*|* · x* have continuous derivatives and their Wronskian vanishes everywhere, yet they are not linearly dependent in any neighborhood of 0.

- Maxime Bôcher observed that if the functions are analytic, then the vanishing of the Wronskian in an interval implies that they are linearly dependent.
^{[3]} - Bôcher (1901) gave several other conditions for the vanishing of the Wronskian to imply linear dependence; for example, if the Wronskian of
*n*functions is identically zero and the*n*Wronskians of*n*– 1 of them do not all vanish at any point then the functions are linearly dependent. - Wolsson (1989a) gave a more general condition that together with the vanishing of the Wronskian implies linear dependence.

Over fields of positive characteristic *p* the Wronskian may vanish even for linearly independent polynomials; for example, the Wronskian of *x*^{p} and 1 is identically 0.

In general, for an th order linear differential equation, if solutions are known, the last one can be determined by using the Wronskian.

Consider the second order differential equation in Lagrange's notation:

where , are known, and y is the unknown function to be found. Let us call the two solutions of the equation and form their Wronskian

Then differentiating and using the fact that obey the above differential equation shows that

Therefore, the Wronskian obeys a simple first order differential equation and can be exactly solved:

where and is a constant.

Now suppose that we know one of the solutions, say . Then, by the definition of the Wronskian, obeys a first order differential equation:

and can be solved exactly (at least in theory).

The method is easily generalized to higher order equations.

For *n* functions of several variables, a **generalized Wronskian** is a determinant of an *n* by *n* matrix with entries *D _{i}*(

The Wronskian was introduced by Józef Hoene-Wroński (1812) and given its current name by Thomas Muir (1882, Chapter XVIII).