Order of differential equation

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Discussion Overview

The discussion revolves around determining the order of a differential equation related to a family of parabolas with a fixed directrix. Participants explore the general equation of such parabolas and the implications of constants involved in the formulation.

Discussion Character

  • Exploratory, Technical explanation, Conceptual clarification

Main Points Raised

  • One participant asks for the order of the differential equation associated with a family of parabolas having a fixed directrix.
  • Another participant inquires about the general equation of this family and notes that differentiating once can eliminate one constant, suggesting that the number of constants corresponds to the order of the equation.
  • A participant expresses difficulty in finding the general equation and requests hints.
  • A later reply provides a hint regarding the definition of a parabola in relation to its focus and directrix, questioning the dimensions and constraints on the focus.
  • Another participant describes the relationship between the vertex, focus, and directrix of a parabola, providing a specific equation and asking about the number of parameters involved.

Areas of Agreement / Disagreement

Participants have not reached a consensus on the general equation or the order of the differential equation, and multiple viewpoints and uncertainties remain regarding the formulation and parameters involved.

Contextual Notes

There are unresolved aspects regarding the assumptions about the dimensions and constraints on the focus, as well as the specific formulation of the general equation.

lizzie
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Find the order of the differential equation of the following curve:
Family of parabolos having fixed directrix.

Thanks for any help.
 
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What does the general equation of such a family look like? How many constants are there in that general formula. Generally speaking you need to differentiate once to eliminate one constant. The number of constants is the order of the equation.
 
I am unable to find the general equation. Please give me some hint.
 
lizzie said:
I am unable to find the general equation. Please give me some hint.

Hi lizzie! :smile:

(how many dimensions is this in? and is the position of the focus constrained in any way?)

Hint: for a fixed focus and directrix, the parabola is all points whose distance from the focus equals its (perpendicular) distance from the directrix. :smile:
 
The vertex of a parabola is exactly half way between the focus and the directrix so a parabola having focus (0, f) and directrix y= -d has vertex at (0, (f-d)/2). The focal distance (distance from the vertex to the focus) is then f- (f-d)/2= (f+d)/2. The equation of such a parabola is y= 2(f+d)x2+ (f-d)/2. How many parameters (constants) does that involve?
 

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