# Documentation

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# ode::wronskian

Wronskian of functions or of a linear homogeneous ordinary differential equation

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## Syntax

ode::wronskian(l, x, <R>)
ode::wronskian(Ly, y(x), <R>)

## Description

ode::wronskian computes the wronskian (determinant) of functions or of a linear homogeneous ordinary differential equation.

ode::wronskian(l, x) returns the wronskian, i.e. the determinant of the wronskian matrix, of the elements of l with respect to x.

ode::wronskian(Ly, y(x)) returns the wronskian of Ly defined as the element w such that , where an - 1 is the coefficient of Ly of degree n - 1 and n the order of Ly.

If the optional argument R is given, then the specified differential ring will be chosen for representing the entries of the wronskian matrix.

## Examples

### Example 1

We compute the wronskian of [2*x^2+1, x*sqrt(1+x^2), y(x)] which is a linear differential equation in y(x):

Ly:=ode::wronskian([2*x^2+1, x*sqrt(1+x^2), y(x)], x)

Ly := numer( normal(Ly) )

And we can check that a basis of solutions of Ly is as expected:

ode::solve(Ly, y(x))

We can also compute the wronskian of Ly, which is, up to a constant, the wronskian of x^2+1 and x*sqrt(x^2+1):

ode::wronskian(Ly, y(x)),
simplify(ode::wronskian([x^2+1/2,x*sqrt(1+x^2)], x))

## Parameters

 l A list of functions of the variable x. Ly A homogeneous linear ordinary differential equation. y(x) The dependent function of Ly. R A differential ring, default is Dom::ExpressionField(id, iszero@normal).

Expression in x.