- #1
Thiendrah
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Is there any exception where I can't use wronskian rule to see if given functions are linearly independent or dependent?
Thanks...
Thanks...
Can Wronskian be used for all?
- Thread starter Thiendrah
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So, to summarize, the Wronskian rule can only be used to determine if functions are linearly independent if they are all solutions to the same linear differential equation and are at least twice differentiable. Additionally, the number of differentials must match the number of functions.
- #2
matematikawan
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You can't use the Wronskian if the functions are not differentiable! 
- #3
HallsofIvy
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Indeed, if the functions in question are not all solutions to the same linear differential equation, then the Wronskian does not help.
So, to use the Wronskian to determine whether two functions are linearly independent they must be twice differrentiable, for three functions, thrice differentiable, etc.
So, to use the Wronskian to determine whether two functions are linearly independent they must be twice differrentiable, for three functions, thrice differentiable, etc.
- #4
Amith2006
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HallsofIvy said:
In order to find the wronskian of n functions, it is enough that they have (n-1) differentials because you will then have the same number of equations as the functions. Sorry for being late to the party!
- #5
HallsofIvy
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Yes, thanks for the correction.
Related to Can Wronskian be used for all?
1. Can the Wronskian be used to determine linear independence for any set of functions?
Yes, the Wronskian can be used to determine linear independence for any set of functions, whether they are polynomials, trigonometric functions, or any other type of function. As long as the functions are continuous and have a finite number of derivatives, the Wronskian can be used to determine their linear independence.
2. Is the Wronskian only applicable to systems of ordinary differential equations?
No, the Wronskian can also be used for partial differential equations and systems of partial differential equations. As long as the functions involved are continuous and have a finite number of derivatives, the Wronskian can be used to determine linear independence.
3. Can the Wronskian be used to solve differential equations?
No, the Wronskian is not used to solve differential equations directly. It is used to determine linear independence, which is a necessary condition for finding the general solution to a system of differential equations. The Wronskian can also be used to verify the correctness of a proposed general solution.
4. Are there any limitations to using the Wronskian?
The Wronskian can only be used when the functions involved are continuous and have a finite number of derivatives. It also requires that the functions are linearly independent, which means they cannot be multiples of each other. Additionally, the Wronskian may not be useful for very large systems of equations, as its computation can become computationally intensive.
5. Can the Wronskian be used for non-numerical functions?
Yes, the Wronskian can be used for both numerical and non-numerical functions. As long as the functions are continuous and have a finite number of derivatives, the Wronskian can be used to determine their linear independence. This includes functions that are defined analytically, symbolically, or graphically.
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