Is the Sign of the Wronskian Arbitrary in Differential Equations?

In summary, the conversation revolved around determining the Wronskian of two solutions to a DE and how the sign of the Wronskian is determined by the ordering of the functions. The person taking the test received points off for having a different sign in their Wronskian result compared to the solution sheet. They realized that the order of the functions is arbitrary and have been taking the easier derivative first. They decided not to argue this point.
  • #1
1MileCrash
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On my DE test, I was asked to determine if two solutions to a DE are fundamental solutions.

So I confirmed they were both solutions, and took the Wronskian, which was nonzero.

I got points marked off, and he put a minus sign in front of my wronskian result.

Isn't the sign of the Wronskian determined by what function I call y1 and what function I call y2 and is thus completely arbitrary?
 
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  • #2
Yes the sign on the Wronskian is determined by the ordering that you assign to the functions. It's a weak excuse, but the grader was probably just comparing your answer with their solution sheet and saw the sign difference and didn't think about why they were different.
 
  • #3
Well, I looked at the test and on the paper the functions were called y1 and y2 already, and in my work I did write W[y1,y2] = my wronskian.

So I guess I can understand. I don't think I'll argue this one. I noticed that the wronskian order is arbitrary early off and have just been taking the easier derivative first.
 

1. What is the Wronskian sign?

The Wronskian sign is a mathematical concept used in linear algebra to determine the linear independence or dependence of a set of functions. It is denoted by W and can be either positive, negative, or zero.

2. Is the Wronskian sign arbitrary?

No, the Wronskian sign is not arbitrary. It depends on the functions being evaluated and can be calculated using specific mathematical formulas.

3. How is the Wronskian sign calculated?

The Wronskian sign is calculated by taking the determinant of the Wronskian matrix, which is formed by arranging the functions and their derivatives in a specific order.

4. What does a positive Wronskian sign indicate?

A positive Wronskian sign indicates that the set of functions is linearly independent, meaning that no function in the set can be expressed as a linear combination of the others.

5. Can the Wronskian sign be used to determine the linear independence of any set of functions?

No, the Wronskian sign can only be used to determine the linear independence of a set of functions when they are solutions to a differential equation. It cannot be used for arbitrary functions.

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