2nd order DE violating theroem 2?

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In summary, the two different solutions y1 = x2 and y2 = x3 of the differential equation x2y'' - 4xy' + 6y = 0 do not contradict Theorem 2 regarding guaranteed uniqueness because the theorem requires continuity, which is not satisfied in this case due to the discontinuity at x = 0. Therefore, the theorem does not apply here and the two solutions are both valid.
  • #1
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Homework Statement


y1 = x2 and y2 = x3 are two different solutions of x2y'' - 4xy' + 6y = 0, both satisfying the initial conditions y(0) = 0 = y'(0). Explain why these facts don't contradict Theorem 2 (with respect to the guaranteed uniqueness).


Homework Equations





The Attempt at a Solution


I have no attempt because I don't understand how it doesn't violate the theorem?
 
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  • #2
Since I have no idea exactly what "theorem 2" is in your textbook (you say "with respect to the guarenteed uniqueness" but there are many theorems about uniqueness) so I can't say what, if any, part of the hypotheses does not hold.

Please state exactly what "theorem 2" says.
 
  • #3
Oh wow, I'm sorry, hadn't even realized that. Theorem 2 is the 2nd order uniqueness - existence theorem - how if there is a solution on interval I satisfying the initial condition, it is the one and only solution to that DE.
 
  • #4
So you are refusing to state exactly what "theorem 2" says? Not much point in continuing this, is there?
 
  • #5
Note that the "solution" to your ODE is this:

y(x) = A*x^2 + B*x^3

*A, B are constants.

and this is the only possible solution.

(as it is a ordinary linear homogeneous equation, its solution is made from any constant multiplicands of the 2 linearly interdependent solution to that ODE.)

Then, there is only a single pair of constants A and B that satisfy a certain IVP.
 
  • #6
HallsofIvy, I gave you a quick description of the theorem, thinking that you would go "Ah, yes, that one" and be able to help me. Did you want exact wording? I'm sorry for the miscommunication.

gomunkul51, thanks for your reply, it was well-worded and clear.

I figured it out, though - the theorem requires continuity and once my equation is put into 'standard form,' I got numbers divided by x.. which obviously means discontinuity where x = 0, so the theorem does not apply here, so it does not violate it.

Thanks!
 
  • #7
mbradar2 said:
HallsofIvy, I gave you a quick description of the theorem, thinking that you would go "Ah, yes, that one" and be able to help me. Did you want exact wording? I'm sorry for the miscommunication.

gomunkul51, thanks for your reply, it was well-worded and clear.

I figured it out, though - the theorem requires continuity and once my equation is put into 'standard form,' I got numbers divided by x.. which obviously means discontinuity where x = 0, so the theorem does not apply here, so it does not violate it.
Yes, that's the key - putting the diff. eqn. in standard form, making it y'' - (4/x)y' + (6/x^2)y = 0, together with the fact that both initial conditions are at x = 0.
 

Related to 2nd order DE violating theroem 2?

1. What is the second-order DE violating theorem 2?

The second-order DE violating theorem 2, also known as the Poincaré-Hopf theorem, is a mathematical theorem that relates the topology of a smooth vector field on a manifold to the singularities of the vector field. It states that if a vector field on a closed, orientable manifold has a finite number of singularities, then the sum of the indices of these singularities is equal to the Euler characteristic of the manifold.

2. How is this theorem used in science?

The second-order DE violating theorem 2 is used in various fields of science, such as physics, engineering, and biology. It is particularly useful in understanding the behavior of dynamical systems, as it allows for the prediction of the number and types of singularities in a given system. This can aid in the analysis and optimization of complex systems.

3. Can you provide an example of how this theorem is applied?

An example of how the second-order DE violating theorem 2 is applied is in the study of fluid dynamics. By considering the vector field of a fluid flow, one can use this theorem to predict the number and types of critical points, or singularities, in the flow. This information can then be used to analyze the stability and behavior of the fluid flow.

4. Are there any limitations to this theorem?

While the second-order DE violating theorem 2 is a powerful tool, it does have some limitations. It only applies to closed, orientable manifolds, and the vector field must be smooth and have a finite number of singularities. Additionally, it does not provide information about the precise locations of the singularities, only their sum and types.

5. How does this theorem relate to other mathematical concepts?

The second-order DE violating theorem 2 is closely related to other mathematical concepts, such as differential equations, topology, and algebraic geometry. It is often used in conjunction with other theorems and techniques, such as the Poincaré index theorem and the Brouwer fixed-point theorem, to analyze and understand complex systems in mathematics and science.

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