Wronskian


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Wronskian

[′vrän·skē·ən]
(mathematics)
An n × n matrix whose i th row is a list of the (i - 1)st derivatives of a set of functions f1, …, fn ; ordinarily used to determine linear independence of solutions of linear homogeneous differential equations.

Wronskian

 

a functional determinant composed of n functions f1(x), f2(x)....,fn(x) and their derivatives up to the order n - 1 inclusive:

The vanishment of the Wrońskian [W(x) = 0] is a necessary and, under certain additional assumptions, a sufficient condition for the linear dependence between the given n functions, differentiated n - 1 times. Based on this, the Wrońskian is used in the theory of linear differential equations. The Wrońskian was introduced by J. Wroński in 1812.

References in periodicals archive ?
It is well-known that f is non-degenerate over C if and only if the Wronskian W = W([f.sub.1], ..., [f.sub.n+1]) of [f.sub.1], ..., [f.sub.n+1] is not identically equal to zero.
In the literature, DWPs have been studied by using various techniques such as the WKB approximation [33, 34], asymptotic iteration method (AIM) [35], and the Wronskian method [36].
Following [8], we introduce the Wronskian of two sequences [{[[psi].sub.1]}.sub.j[member of]N], and [{[[psi].sub.2]}.sub.j[member of]N],
Many methods have been developed to find the explicit solutions of nonlinear evolution equations; example of such methods are the first integral method [9], Jacobi elliptic function method [10], Hirota bilinear method [11], Wronskian determinant technique [12], F-expansion method [13], Darboux Transformations [14], Backlund transformation method [6], Miura transformation [15], homotopy perturbation method [16], and Adomian decomposition method [17].
hence [[g.sub.2], [y.sub.m]] = 0, and by the Wronskian identity, we get
where W[[phi](x, [lambda]), [psi](x, [lambda])] is Wronskian of the vector solutions [phi](x, [lambda]) and [psi](x, [lambda]).
[28] constructed its Wronskian and Grammian solutions.
Let f and g be a-differentiable; the fractional Wronskian function is defined by
Let W(f g; x) = f(x) g'(x) - f'(x)g(x) denote the Wronskian of functions f(x) and g(x).
and [W.sub.n] is the Wronskian of Whittaker functions [16]