Wronskian - Revision history

Khanh at 04:20, 6 November 2021

2021-11-06T04:20:18Z

Lila at 21:48, 14 October 2021

2021-10-14T21:48:59Z

Johnraymond.yanez at 14:34, 12 August 2020

2020-08-12T14:34:48Z

Johnraymond.yanez: Created page with "[https://tutorial.math.lamar.edu/Classes/DE/Wronskian.aspx]. Produced by Paul Dawkins, Lamar University"

2020-08-12T14:34:20Z

Created page with "[https://tutorial.math.lamar.edu/Classes/DE/Wronskian.aspx]. Produced by Paul Dawkins, Lamar University"

New page

[https://tutorial.math.lamar.edu/Classes/DE/Wronskian.aspx]. Produced by Paul Dawkins, Lamar University

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 ==Resources==
 * [https://tutorial.math.lamar.edu/Classes/DE/Wronskian.aspx Notes on the Wronskian]. Produced by Paul Dawkins, Lamar University
-* [https://en.wikibooks.org/wiki/Ordinary_Differential_Equations/Linear_Equations Linear Equations], Wikibooks: Ordinary Differential Equations

← Older revision		Revision as of 21:48, 14 October 2021
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−	[https://tutorial.math.lamar.edu/Classes/DE/Wronskian.aspx Notes on the Wronskian]. Produced by Paul Dawkins, Lamar University	+	==Wronskian==
		+
		+	Let L(y)=0 be a homogeneous equation of degree n, and let <math>y_1,y_2,...,y_n</math> be linearly independent solutions of this equation.
		+
		+	The general solution is then <math>y=C_1y_1+C_2y_2+...+C_ny_n</math>.
		+
		+	Suppose instead that the solutions are not linearly independent. Then there exists values for <math>C_1,C_2,...,C_n</math> not all zero such that <math>C_1y_1+C_2y_2+...+C_ny_n</math>=0.
		+
		+	Then the following equations also hold true:
		+
		+	<math>C_1y_1'+C_2y_2'+...+C_ny_n'=0</math><br />
		+	<math>C_1y_1''+C_2y_2''+...+C_ny_n''=0</math><br />
		+	...<br />
		+	<math>C_1y_1^{(n-1)}+C_2y_2^{(n-1)}+...+C_ny_n^{(n-1)}=0</math>
		+
		+	When there is to be a non-trivial solution to this system of homogeneous linear equations, the column vectors corresponding to each coefficient are linearly dependent. This is equivalent to saying that the following determinant, called the '''Wronskian''' is 0:
		+
		+	<math>
		+	\begin{Vmatrix}
		+	y_1 & y_2 & ... & y_n \\
		+	y_1' & y_2' & ... & y_n' \\
		+	y_1'' & y_2'' & ... & y_n'' \\
		+	... & ... & ... & ... \\
		+	y_1^{(n-1)} & y_2^{(n-1)} & ... & y_n^{(n-1)} \\
		+	\end{Vmatrix} = 0
		+	</math>
		+
		+	Thus, if the solutions are dependent, then their Wronskian is equal to 0, and conversely, and conversely if the Wronskian is equal to 0, then the columns of that matrix must be linearly dependent, indicating that the solutions are linearly dependent.
		+
		+	==Resources==
		+	* [https://tutorial.math.lamar.edu/Classes/DE/Wronskian.aspx Notes on the Wronskian]. Produced by Paul Dawkins, Lamar University
		+	* [https://en.wikibooks.org/wiki/Ordinary_Differential_Equations/Linear_Equations Linear Equations], Wikibooks: Ordinary Differential Equations

← Older revision		Revision as of 14:34, 12 August 2020
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−	[https://tutorial.math.lamar.edu/Classes/DE/Wronskian.aspx]. Produced by Paul Dawkins, Lamar University	+	[https://tutorial.math.lamar.edu/Classes/DE/Wronskian.aspx Notes on the Wronskian]. Produced by Paul Dawkins, Lamar University