Wronskian
In mathematics, the Wronskian of n differentiable functions is the determinant formed with the functions and their derivatives up to order. It was introduced in 1812 by the Polish mathematician Józef Wroński, and is used in the study of differential equations, where it can sometimes show the linear independence of a set of solutions.
Definition
The Wrońskian of two differentiable functions and is.More generally, for real- or complex-valued functions, which are times differentiable on an interval, the Wronskian is a function on defined by
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 are solutions of a linear differential equation, the Wrońskian can be found explicitly using Abel's identity, even if the functions are not known explicitly.
The Wronskian and linear independence
If the functions are linearly dependent, then so are the columns of the Wrońskian, and the Wrońskian vanishes. Thus, one may show that a set of differentiable functions is linearly independent on an interval by showing that their Wrońskian does not vanish identically. It may, however, vanish at isolated points.A common misconception is that everywhere implies linear dependence. pointed out that the functions and have continuous derivatives and their Wrońskian vanishes everywhere, yet they are not linearly dependent in any neighborhood of. There are several extra conditions which combine with vanishing of the Wronskian in an interval to imply linear dependence.
- Maxime Bôcher observed that if the functions are analytic, then the vanishing of the Wrońskian in an interval implies that they are linearly dependent.
- gave several other conditions for the vanishing of the Wrońskian to imply linear dependence; for example, if the Wrońskian of functions is identically zero and the Wrońskians of of them do not all vanish at any point then the functions are linearly dependent.
- gave a more general condition that together with the vanishing of the Wronskian implies linear dependence.
Application to linear differential equations
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 Wrońskian, obeys a first order differential equation:
and can be solved exactly.
The method is easily generalized to higher order equations.
The relationship between the Wronskian and linear independence can also be strengthened in the context of a differential equation. If we have linearly independent functions that are all solutions of the same monic th-order homogeneous-linear ordinary differential equation on some interval, then their Wronskian is zero nowhere on. Thus, counterexamples like and or even and are ruled out; neither pair can consist of solutions to the same second-order differential equation of this type.