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January 28th, 2019, 04:28 AM   #1
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Wronskian application

What is Wronskian used for?
Here is the page of definition, but I need a simple example.

Last edited by skipjack; January 28th, 2019 at 09:26 PM.
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January 28th, 2019, 08:16 PM   #2
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There is a section on that wiki page involving ODEs as an example.
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January 29th, 2019, 02:08 AM   #3
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Wronskian shows whether the solutions are linearly independent.
If they are linearly independent the page says, $\displaystyle y_1 =xy_2$ .
Why not $\displaystyle y_1=(ax+b)y_2$ ?
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January 29th, 2019, 07:36 PM   #4
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The only application I know of is an easy proof that some collection of functions are linearly independent. Namely, if $W(f_1,\dots,f_n) \neq 0$ on some domain, $D$, then the set of functions $\{f_1,\dots,f_n\}$ restricted to $D$ form a linearly independent set.

Linear independence does not mean $y_1 = xy_2$ nor does it mean $y_1 = (ax + b)y_2$. I don't know where you got either of these formulas. The definition says that if $V$ is a (finite-dimensional) vector space, then $\{v_1,\dots,v_n\} \subset V$ are linearly independent if whenever $a_1v_1 + \dots + a_nv_n = 0$ you must have $a_j = 0$ for every $j$. In English, if a linear combination of linearly independent vectors is zero, then every coefficient is zero.

In the case of the Wronskian, the vector space is the space of functions which are $n$ times differentiable so that this determinant makes sense. This is a standard example of a vector space which you should have seen in a first linear algebra class.
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