Wronskian

In mathematics, the Wronskian of n differentiable functions is the determinant formed with the functions and their derivatives up to order Template:Nowrap. 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 Template:Math and Template:Math is <math> W(f,g)=f g' - g f' </math>.

More generally, for Template:Math real- or complex-valued functions Template:Math, which are Template:Math times differentiable on an interval Template:Math, the Wronskian <math> W(f_1,\ldots,f_n) </math> is a function on <math> x\in I </math> defined by <math display="block"> W(f_1, \ldots, f_n) (x)= \det \begin{bmatrix} 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{bmatrix}. </math>

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 <math> (n-1)^{\text{th}} </math> derivative, thus forming a square matrix.

When the functions Template:Math are solutions of a linear differential equation, the Wrońskian can be found explicitly using Abel's identity, even if the functions Template:Math are not known explicitly. (See below.)

The Wronskian and linear independence

If the functions Template:Math are linearly dependent, then so are the columns of the Wrońskian (since differentiation is a linear operation), 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.<ref name="BenderAndOrszag">Template:Citation</ref>

A common misconception is that Template:Math everywhere implies linear dependence. Template:Harvtxt pointed out that the functions Template:Math and Template:Math have continuous derivatives and their Wrońskian vanishes everywhere, yet they are not linearly dependent in any neighborhood of Template:Math.Template:Efn 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.<ref name="PeanoOnWronskians-BocherAnalytic" />
• Template:Harvtxt gave several other conditions for the vanishing of the Wrońskian to imply linear dependence; for example, if the Wrońskian of Template:Math functions is identically zero and the Template:Math Wrońskians of Template:Math of them do not all vanish at any point then the functions are linearly dependent.
• Template:Harvtxt gave a more general condition that together with the vanishing of the Wronskian implies linear dependence.

Over fields of positive characteristic Template:Math the Wronskian may vanish even for linearly independent polynomials; for example, the Wronskian of Template:Math and 1 is identically 0.

Application to linear differential equations

In general, for an <math>n</math>th order linear differential equation, if <math> (n-1) </math> solutions are known, the last one can be determined by using the Wronskian.

Consider the second order differential equation in Lagrange's notation: <math display="block">y = a(x)y' + b(x)y</math> where <math>a(x)</math>, <math>b(x)</math> are known, and y is the unknown function to be found. Let us call <math> y_1, y_2 </math> the two solutions of the equation and form their Wronskian <math display="block">W(x) = y_1 y'_2 - y_2 y'_1</math>

Then differentiating <math> W(x) </math> and using the fact that <math> y_i </math> obey the above differential equation shows that <math display="block">W'(x) = a(x) W(x)</math>

Therefore, the Wronskian obeys a simple first order differential equation and can be exactly solved: <math display="block">W(x) = C~e^{A(x)}</math> where <math> A'(x)=a(x) </math> and <math>C</math> is a constant.

Now suppose that we know one of the solutions, say <math> y_2 </math>. Then, by the definition of the Wrońskian, <math> y_1 </math> obeys a first order differential equation: <math display="block"> y'_1 -\frac{y'_2}{y_2} y_1 = -W(x)/y_2</math> and can be solved exactly (at least in theory).

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 <math>n</math> linearly independent functions that are all solutions of the same monic <math>n</math>th-order homogeneous-linear ordinary differential equation <math>y^{(n)}+Ly=0</math> (where <math>L</math> is a linear differential operator with respect to <math>x</math> of order less than <math>n</math>) on some interval <math>I</math>, then their Wronskian is zero nowhere on <math>I</math>. Thus, counterexamples like <math>x^2</math> and <math>x{|x|}</math> (whose Wronskian is zero everywhere) or even <math>x^2</math> and <math>1</math> (whose Wronskian <math>2x</math> is zero somewhere) are ruled out; neither pair can consist of solutions to the same second-order differential equation of this type. (It's true that <math>x^2</math> and <math>1</math> are both solutions to the same third-order differential equation <math>y^{(3)}=0</math>. But the Wronskian <math>-2</math> of the three independent solutions <math>x^2</math>, <math>x</math>, and <math>1</math> is nowhere zero.)

Generalized Wrońskians

For Template:Math functions of several variables, a generalized Wronskian is a determinant of an Template:Math by Template:Math matrix with entries Template:Math (with Template:Math), where each Template:Math is some constant coefficient linear partial differential operator of order Template:Math. If the functions are linearly dependent then all generalized Wronskians vanish. As in the single variable case the converse is not true in general: if all generalized Wronskians vanish, this does not imply that the functions are linearly dependent. However, the converse is true in many special cases. For example, if the functions are polynomials and all generalized Wronskians vanish, then the functions are linearly dependent. Roth used this result about generalized Wronskians in his proof of Roth's theorem. For more general conditions under which the converse is valid see Template:Harvtxt.

History

The Wrońskian was introduced by Template:Harvs and given its current name by Template:Harvs.

See also

• Variation of parameters
• Moore matrix, analogous to the Wrońskian with differentiation replaced by the Frobenius endomorphism over a finite field.
• Alternant matrix
• Vandermonde matrix

References

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