Integrating Factors: Spotting D.E. Need for Factor

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Discussion Overview

The discussion revolves around identifying when a first-order differential equation (D.E.) requires an integrating factor. Participants explore the conditions under which integrating factors are applicable and the challenges associated with finding them.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant suggests that a D.E. needs an integrating factor if it can be expressed in the form dy/dx + Py = Q, where P and Q are functions of x.
  • Another participant states that if a D.E. is not already exact, it requires an integrating factor, but finding that factor can be challenging.
  • A different viewpoint indicates that every first-order differential equation has an integrating factor, but the difficulty lies in determining what that factor is.
  • It is mentioned that for linear first-order differential equations, there exists a simple formula for the integrating factor.
  • One participant references the concept of integration by parts and the product rule as related ideas in the context of integrating factors.

Areas of Agreement / Disagreement

Participants express differing views on the necessity and definition of integrating factors, with some asserting that all first-order D.E.s require them while others emphasize the difficulty in finding them. The discussion remains unresolved regarding the clarity of what constitutes a need for an integrating factor.

Contextual Notes

Participants do not fully clarify the implications of "exact" equations or the specific conditions under which integrating factors are deemed necessary, leaving some assumptions unaddressed.

asdf1
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how do you spot that a D.E. needs an integrating factor, besides experience?
 
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If you're talking about 1st order O.D.Es, it's if you can get it of the form:

[tex]\frac{dy}{dx} + Py = Q[/tex]

Where P and Q are functions of x.
 
thanks! :)
 
If it's not already exact, it needs an integrating factor!

The hard part is finding that integrating factor.
 
it is just an extension of integration by parts, i.e. trying to use the product rule. "there are no new ideas under the sun".
 
I'm not sure what you mean by "needs" an integrating factor. Every first order differential equation has an integrating factor but it may be very difficult to find. That's what I meant when I said before "If it's not already exact, it needs an integrating factor!"

If a first order differential equation is linear then there is a simple formula for the integrating factor. I presume that was what Zurtex meant.
 
thank you!
 

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