Is This ODE a Bernoulli Equation and Can It Be Solved with Substitution?

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Discussion Overview

The discussion revolves around the classification and potential solution methods for a specific ordinary differential equation (ODE). Participants explore whether the ODE can be categorized as a Bernoulli equation and if it can be solved using substitution techniques.

Discussion Character

  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant suggests that the ODE resembles a Bernoulli equation with P(x)=0 and inquires about the applicability of Bernoulli methodology using the substitution u=y^(1-a) with a=2.
  • Another participant claims that the ODE is separable and provides a method to manipulate it by dividing both sides by y^2 and multiplying by dx.
  • A third participant confirms that the ODE is a first-order, non-linear differential equation, noting that it is first-order due to the highest derivative being first and non-linear because the dependent variable is not first-degree. They also mention that separation of variables is a common technique in differential equation textbooks.

Areas of Agreement / Disagreement

Participants express differing views on the classification of the ODE, with some supporting the idea of it being a Bernoulli equation while others emphasize its separability. The discussion remains unresolved regarding the best approach to solve the ODE.

Contextual Notes

There are limitations in the discussion regarding the assumptions made about the form of the ODE and the specific conditions under which the proposed methods may be applicable. The exact nature of the ODE is not fully defined in the posts.

Houeto
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upload_2016-7-15_16-38-10.png

consider ODE :
upload_2016-7-15_16-38-46.png

Show that the solution to this ODE is:
upload_2016-7-15_16-40-23.png


Can someone tell what kind of ODE is it?I thought,it's on the form of Bernoulli ODE with P(x)=0.Is it possible to still solve it by using Bernoulli Methodology?I mean by substituting u=y^1-a with a=2?

Thanks
 
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It's separable. Divide both sides by ##y^{2}##, multiply both sides by dx, and you'll see what I mean.
 
Houeto said:
Can someone tell what kind of ODE is it?
The DE is a first order, non-linear differential equation. It's first order, since the highest derivative is a first derivative. It's nonlinear, since the dependent variable is not first-degree.

As Twigg points out, it turns out to be separable, so you can manipulate it to get y and dy on one side and x and dx on the other. Solving DEs by separation is one of the first techniques presented in most diff. equation textbooks.
 
Thanks Guys!
 

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