Wronskian: Difference between revisions

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As such, the Wronskian is either always 0 (when C is 0 and the functions are linearly dependent) or never 0 (when the C is nonzero and the functions are linearly independent).
As such, the Wronskian is either always 0 (when C is 0 and the functions are linearly dependent) or never 0 (when the C is nonzero and the functions are linearly independent).
= Systems =
General solutions to systems of equations are expressed as the following:
<math>
c_1\vec{x}^{(1)}+\ldots+c_k\vec{x}^{(k)}=0
</math>
For them, the Wronskian is defined to be the determinant of X. We have another clause of [[Abel's theorem]] and the following
<math>
W=Cexp[\int trP(t)dt]
</math>
We can then draw a similar conclusion as above: the Wronskian is either always 0 or never 0.

Latest revision as of 14:47, 10 June 2024

The Wronskian of n equations is the determinant of the following matrix.

It can be used to check if solutions of differential equations are linearly independent (when the Wronskian is nonzero).

Properties

For the Wronskians of solutions of linear differential equations,

As such, the Wronskian is either always 0 (when C is 0 and the functions are linearly dependent) or never 0 (when the C is nonzero and the functions are linearly independent).

Systems

General solutions to systems of equations are expressed as the following:

For them, the Wronskian is defined to be the determinant of X. We have another clause of Abel's theorem and the following

We can then draw a similar conclusion as above: the Wronskian is either always 0 or never 0.