Wronskian Properties and Variation of Parameters

[kasse]

What's the wronskian of x^2 and x^-2?

I've found a basis of solutions to a non-homogeneous 2nd order ODE and want to find a particuler solution using variation of parameters.

 

[genneth]

Use the definition: http://en.wikipedia.org/wiki/Wronskian

 

[kasse]

But how do I know which function is y1 and y2?

 

[genneth]

Does it matter? What happens to the determinant of a matrix when you transpose two columns or rows?

 

[kasse]

For instance:

W(e^2x, e^x)= (-e^3x)

and

W(e^x, e^2x)= e^3x

Different wronskians...

 

[HallsofIvy]

Yes, you get different signs. Now is that diffence in sign important in your application?

 

[kasse]

Variation of parameters. I think its important.

 

[genneth]

Variation of parameters. I think its important.

What you will find is that everywhere the Wronskian is use, there's an accompanying minus sign. So if you switch the two solution about, yes the Wronskian changes sign, but so does that the order that the two solutions appear around the minus sign. Thus, there is no difference in the end.