Differential equations - Decidability and Complexity

In summary, the conversation discusses linear differential equations with polynomial coefficients and the related problems of finding solutions, determining linear independence, and the decidability and complexity of these problems. There is also a question about the existence of an algorithm and the connection to the 10th problem of Hilbert.
  • #1
mathmari
Gold Member
MHB
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Hey! :eek:

Is someone familiar with the following?

We have linear differential equations with polynomial coefficients depending on x.

$a_n(x)y^{(n)}+ \dots a_1(x)y^{(1)}+a_0(x)y^{(0)}=b(x)$

There are problems like if there are solutions, if the solutions are linear independent and so on and we are looking for the decidability and the complexity.
 
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  • #2
Hi,

I'm not too much familiar with this questions, but ... don't you need an algorithm to talk about the complexity? Or are you asking about the existence of a polynomial time algorithm?
 
  • #3
First of all, I am asking if someone is familiar with the decidability of such problems.

Are you familiar with that?
 
  • #4
One such problem is the following:

View attachment 4527

Do you maybe know where I can get more information?
 

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  • #5
Is this related to the 10th problem of Hilbert?
 

1. What is the difference between decidability and complexity in differential equations?

Decidability refers to the ability to determine whether a given differential equation has a solution, while complexity refers to the level of difficulty in finding that solution. In other words, decidability is a binary concept (yes or no), while complexity is a measure of the amount of resources (time, memory, etc.) required to find a solution.

2. How do we determine if a differential equation is decidable?

A differential equation is decidable if there exists an algorithm that can determine whether it has a solution or not. This algorithm must work for all possible inputs and terminate in a finite amount of time.

3. Can all differential equations be solved using algorithms?

No, not all differential equations are decidable. In fact, most real-world differential equations are undecidable, meaning there is no algorithm that can determine whether they have a solution or not.

4. How does the complexity of a differential equation affect its solvability?

The complexity of a differential equation can greatly impact its solvability. The more complex a differential equation is, the more difficult and resource-intensive it is to find a solution. In some cases, the complexity may be so high that it is impossible to find a solution within a reasonable amount of time.

5. Are there any techniques for dealing with undecidable differential equations?

Yes, there are several techniques that can be used to approximate solutions for undecidable differential equations. These include numerical methods, perturbation methods, and asymptotic methods. However, these techniques may not always provide accurate solutions and should be used with caution.

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