Wronskian - Revision history

Rice at 14:47, 10 June 2024

2024-06-10T14:47:45Z

← Older revision		Revision as of 14:47, 10 June 2024
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	W=Cexp[\int trP(t)dt]		W=Cexp[\int trP(t)dt]
	</math>		</math>

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

Rice: /* Systems */

2024-06-10T14:46:16Z

Systems

← Older revision		Revision as of 14:46, 10 June 2024
Line 32:		Line 32:
	</math>		</math>

	For them, the Wronskian is defined to be the determinant of X.		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>

Rice at 14:37, 10 June 2024

2024-06-10T14:37:47Z

← Older revision		Revision as of 14:37, 10 June 2024
Line 24:		Line 24:

	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.

Rice at 22:50, 21 May 2024

2024-05-21T22:50:18Z

@@ Line 3: / Line 3: @@
 <math>
-\begin{vmatrix}
+W(y_1,y_2)(x)=\begin{vmatrix}
 f_1(x) & f_2(x) & \cdots & f_n(x) \\
 f_1'(x) & f_2'(x) & \cdots & f_n'(x) \\
@@ Line 12: / Line 12: @@
 It can be used to check if solutions of differential equations are linearly independent (when the Wronskian is nonzero).

Rice at 22:28, 21 May 2024

2024-05-21T22:28:43Z

← Older revision		Revision as of 22:28, 21 May 2024
Line 1:		Line 1:
	[[Category:Differential Equations]]		[[Category:Differential Equations]]
	The '''Wronskian''' of n equations is the determinant of the following matrix		The '''Wronskian''' of n equations is the determinant of the following matrix.

	<math>		<math>
Line 10:		Line 10:
	\end{vmatrix}		\end{vmatrix}
	</math>		</math>

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

Rice: Created page with "Category:Differential Equations The '''Wronskian''' of n equations is the determinant of the following matrix $\begin{vmatrix} f_1(x) & f_2(x) & \cdots & f_n(x) \\ f_1'(x) & f_2'(x) & \cdots & f_n'(x) \\ \vdots & \vdots & \ddots & \vdots \\ f_1^{(n-1)}(x) & f_2^{(n-1)}(x) & \cdots & f_n^{(n-1)}(x) \end{vmatrix}$ "

2024-05-19T19:49:21Z

Created page with "Category:Differential Equations The '''Wronskian''' of n equations is the determinant of the following matrix <math> \begin{vmatrix} f_1(x) & f_2(x) & \cdots & f_n(x) \\ f_1'(x) & f_2'(x) & \cdots & f_n'(x) \\ \vdots & \vdots & \ddots & \vdots \\ f_1^{(n-1)}(x) & f_2^{(n-1)}(x) & \cdots & f_n^{(n-1)}(x) \end{vmatrix} </math>"

New page

[[Category:Differential Equations]]
The '''Wronskian''' of n equations is the determinant of the following matrix

<math>
\begin{vmatrix}
f_1(x) & f_2(x) & \cdots & f_n(x) \\
f_1'(x) & f_2'(x) & \cdots & f_n'(x) \\
\vdots & \vdots & \ddots & \vdots \\
f_1^{(n-1)}(x) & f_2^{(n-1)}(x) & \cdots & f_n^{(n-1)}(x)
\end{vmatrix}
</math>