Question

Can someone explain how the wronskian works when dealing with higher order differential equations? (Please keep in mind that I took linear algebra a while ago, so the matrix stuff is kind of new to me)

Answer 1

For variation of parameters that is...

Answer 2

@electrokid

Answer 3

If the determinant is non zero the columns are linearly independent.

Answer 4

So I assume in this context it would mean you have a basis for the solution set.

Answer 5

I know that much from 2nd order de

Answer 6

Yes.

Answer 7

@amistre64

Answer 8

It is just a tool to check for linear independence. I don't think there is anything deeper unless there is something I do not know about the wronskian.

Answer 9

Well...from the examples that I see..it's being used to find a second solution.

Answer 10

Yes, because if you have one solution you are basically finding the second solution by finding a solution that will lead to a non-zero wronskian.

Answer 11

the second solution comes from the application of Abel's Identity
http://bit.ly/ZRs6Rj

Answer 12

Haha funny intro ty

Answer 13

Is thathe only way?

Answer 14

Yes. this then leads to a method called the "Mathod of Variation os Parameters" where, you can find the solutions of an "n^th" oder DE straight from the Wronskian. This method 'cheats' by starting with the fact that the Wronskian cannot have both "0" and "non-zero" values for a variable.

Answer 15

It was Wronski's idea to test for dependence (IT DOES NOT, though). But Abel's theorem extends it such that it'd provide INDEPENDENT solutions for a DE.
Then by corollaries and simple extensions, came the application to get the solution for an n^th order homo and in-homogeneous DE. http://bit.ly/ZRw4tc

Answer 16

this "tool" greatly simplifies the analysis of any DE.

Answer 17

the development of the concept of MATRICES was a great impetus in development of Scientific Mathematics

Answer 18

Ty for all the great info...

Answer 19

you are welcome.

Answer 20

Quick q from this example: www.utdallas.edu/dept/abp/PDF.../DE.../VariationofParametersII.pdf

Answer 21

What does the cofactor expansion mean?

Answer 22

Matrices have co-factors .. expansion means finding the determinant..
you probably implygin the part hwere Cramer's Rule is applied. (I could not find that file)

Answer 23

Put that link into google

Answer 24

It should be the 1st one

Answer 25

yes. they imply, "find the determinant" of that matrix = Wronskian

Answer 26

Why are they using the 1st column?

Answer 27

Easiest?

Answer 28

no. this is according to Cramer's rule for solution of a set of linear equations. wherein you have to find determinants of the "system" and with each respective variable.

Answer 29

you have too use all the columns to find a determinant.
The easiest is to go row-wise

Answer 30

Ty!