Differential equation y/(x^2+y^2)

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Discussion Overview

The discussion revolves around the differential equation given by $\dfrac{dy}{dx}=\dfrac{y}{x^2+y^2}$. Participants are exploring the correctness of this equation and considering alternative forms, as well as discussing potential transformations to solve it. The scope includes mathematical reasoning and problem-solving approaches related to differential equations.

Discussion Character

  • Exploratory
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant questions the correctness of the original differential equation, suggesting it might instead be $\dfrac{dy}{dx}=\dfrac{y^2}{x^2+y^2}$, which they believe would simplify the problem due to its homogeneity.
  • Another participant emphasizes the importance of showing progress when asking for help, indicating that understanding where one is stuck can lead to better assistance.
  • A suggestion is made to apply a transformation involving $x = -\dfrac{r s'}{s}$ and $y = r$, prompting further exploration of the equation's behavior under this change.

Areas of Agreement / Disagreement

Participants express uncertainty regarding the correct form of the differential equation, with competing views on whether the original or the alternative form is accurate. The discussion remains unresolved as no consensus is reached on the correct equation.

Contextual Notes

There are limitations regarding the assumptions made about the differential equation's form, and the implications of the proposed transformation are not fully explored. The discussion does not clarify the mathematical steps involved in verifying the correctness of the equations.

Nikolas7
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Can you advice the changes in this diff equation:

$\d{y}{x}=\dfrac{y}{x^2+y^2}$
 
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Hi Nikolas7 and happy new year, :)

We ask that a new question is posted in a new thread rather than tagged at the end of an existing thread (http://mathhelpboards.com/rules/).
That's why I have moved your post to a new thread.

We also ask that our users show their progress (work thus far or thoughts on how to begin) when posting questions. This way our helpers can see where you are stuck or may be going astray and will be able to post the best help possible without potentially making a suggestion which you have already tried, which would waste your time and that of the helper.

Can you post what you have done so far?
 
Nikolas7 said:
Can you advice the changes in this diff equation:

$\d{y}{x}=\dfrac{y}{x^2+y^2}$

I am wondering if this is the correct DE. Are you sure it's not $\displaystyle \begin{align*} \frac{\mathrm{d}y}{\mathrm{d}x} = \frac{y^2}{x^2 + y^2} \end{align*}$?
 
Prove It said:
I am wondering if this is the correct DE. Are you sure it's not $\displaystyle \begin{align*} \frac{\mathrm{d}y}{\mathrm{d}x} = \frac{y^2}{x^2 + y^2} \end{align*}$?
This would be wonderful if that was true b/c the equation is homogenous. With the right side numerator of power one, the problem is a little more difficult!
 
Make the transformation

$$x = - \dfrac{r s'}{s},\;\;\; y = r$$

where $$s = s(r)$$ and see where that takes you.
 

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