Transforming a 1st Order PDE into a 2nd Order PDE: A Simple Example

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Discussion Overview

The discussion revolves around transforming a given equation of an ellipsoid into a partial differential equation (PDE). Participants explore the process of deriving a PDE from the implicit equation and the ambiguity in the question regarding the order of the PDE and the form of the solution.

Discussion Character

  • Exploratory
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant requests to form the differential equation for the ellipsoid equation, but the method to derive a PDE from it is unclear to others.
  • Another participant suggests that the task is to find the PDE whose solution corresponds to the implicit equation of the ellipsoid.
  • A participant mentions that deriving a first-order PDE again can yield a second-order PDE, indicating a potential method for transformation.
  • There is a suggestion to consider simpler examples, such as the function y(x) = x^2, to illustrate the concept of multiple differential equations having the same solution.
  • Participants express uncertainty about whether to use parametric or explicit forms of the equations in their derivations.

Areas of Agreement / Disagreement

Participants do not reach a consensus on how to derive the PDE from the given equation, and there are multiple interpretations of the task, indicating ongoing disagreement and uncertainty.

Contextual Notes

The discussion highlights ambiguities in the original question regarding the form of the PDE and the assumptions about the type of equations to be used, as well as the potential for multiple valid approaches.

dvs77
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form the differential equation for the follwing equation
(x^2)/a^2 +(y^2)/b^2+(Z^2)/c^2 =1
 
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Just to keep us busy or what?

Show us your work!
 
I read that as:

Form the differential equation for the following equation:

[tex]\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1[/tex]

I don't get it...how do you get a PDE out of that?
 
No,find the PDE whose solution is the ellipsoid's implicit equation...

Daniel.
 
I see, ok. That wasn't really clear before.
 
I mean u get a First order or second order partial differential equation
 
Well for this last question : suppose you have a 1st order PDE, then just derive again and you get a 2nd order one...

The question is a bit ambigous...let's take a simpler exemple, nonparametric :

y(x)=x^2

Then there are an infinity of differential equation having that solution :

y'=2Sqrt(y)

y''=y'/Sqrt(y)

aso...

Moreover I don't know if you want the parametric equation or the explicit version...(ie. x=x(theta,phi) or x=x(y,z)...)

Just plug in this in your equation and differentiate...
 

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