Returns the Wronskian of the provided functions, differentiating with
respect to the given variable. If no variable is provided,
diff(f) is called for each function f.
wronskian(f1,...,fn, x) returns the Wronskian of f1,...,fn, with
derivatives taken with respect to x.
wronskian(f1,...,fn) returns the Wronskian of f1,...,fn where
k'th derivatives are computed by doing `.derivative(k)' on each
function.
The Wronskian of a list of functions is a determinant of derivatives.
The nth row (starting from 0) is a list of the nth derivatives of the
given functions.
For two functions:
| f g |
W(f, g) = det| | = f*g' - g*f'.
| f' g' |
EXAMPLES:
sage: wronskian(e^x, x^2)
2*x*e^x - x^2*e^x
sage: var('x, y'); wronskian(x*y, log(x), x)
(x, y)
y - log(x)*y
If your functions are in a list, you can use `*' to turn them into
arguments to wronskian():
sage: wronskian(*[x^k for k in range(1, 5)])
12*x^4
If you want to use 'x' as one of the functions in the Wronskian,
you can't put it last or it will be interpreted as the variable
with respect to which we differentiate. There are several ways to
get around this.
Two-by-two Wronskian of sin(x) and e^x:
sage: wronskian(sin(x), e^x, x)
e^x*sin(x) - e^x*cos(x)
Three-by-three Wronskian of sin(x), e^x, and x:
sage: wronskian(sin(x), cos(x), x+0)
x*(-sin(x)^2 - cos(x)^2)
Or don't put x last:
sage: wronskian(x, sin(x), e^x)
x*(e^x*sin(x) + e^x*cos(x)) - 2*e^x*sin(x)
Example where one of the functions is constant:
sage: wronskian(1, e^(-x), e^(2*x))
-6*e^x
NOTES:
http://en.wikipedia.org/wiki/Wronskian
http://planetmath.org/encyclopedia/WronskianDeterminant.html
AUTHORS:
- Dan Drake (2008-03-12)
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