Wronskian of functions or of a linear homogeneous ordinary differential equation
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ode::wronskian(l
,x
, <R
>) ode::wronskian(Ly
, y(x
), <R
>)
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 a_{n 
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.
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))

A list of functions of the variable 

A homogeneous linear ordinary differential equation. 

The dependent function of 

A differential ring, default is 
Expression in x
.