Non-linear Differential Equation - Pulling my Hair

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Discussion Overview

The discussion revolves around a non-linear differential equation presented by a participant, specifically the equation (y'')^2 - xy'' + y' = 0. Participants explore various methods and substitutions to tackle the equation, including potential transformations and classifications.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested, Mathematical reasoning

Main Points Raised

  • One participant expresses frustration with the non-linear differential equation and seeks assistance in solving it.
  • Another participant suggests that the equation can be transformed into a common non-linear form by letting v = y', leading to v = xv' + f(v').
  • A different participant proposes a substitution involving trigonometric functions, specifically x = rcost and 4v = (rsint)^2.
  • Another participant identifies the equation as a Clairaut's ODE and provides a general solution in the form of y = a x² - 4a²x + b, where a and b are constants.
  • The original poster acknowledges the suggestions and expresses a desire to explore the idea of Clairaut's ODE further, mentioning a lack of familiarity with factoring techniques.

Areas of Agreement / Disagreement

Participants present multiple approaches and transformations, indicating a lack of consensus on a single method for solving the equation. The discussion remains open-ended with various suggestions being explored.

Contextual Notes

Some assumptions regarding the applicability of certain methods or transformations are not explicitly stated, and the discussion does not resolve the mathematical steps involved in the proposed solutions.

jkent
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Non-linear Differential Equation - Pulling my Hair !

Hi,
What seems like a simple problem could be going abit better. Any ideas would be sincerely appreciated.

(y'')^2 -xy'' + y' = 0
The squared term is causing me grief !
If I set say v = y' , that still leaves me with the squared term.
(v')^2 - xv' + v =0.

Sorry .. I must be missing something. This looks remarkably quadratic - but its not triggering anything for me right now.

Ideas please and thank you !

J. Kent.
 
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When you let v=y', then we get:

[tex]v=xv'+f(v')[/tex]

That's a common non-linear equation. Wanna' look for it?
 
Last edited:


Try substituting x=rcost, 4v=(rsint)^2.
 


That's a common non-linear equation
A clue : a Clairaut's ODE
Finally:
y = a x²-4a²x+b
a , b = constants
 


Good ideas all. I'll poke at this abit more. Claurauts ODE - hasn't made my list so far - but it clearly needs to. I was told that factoring will work. I just don't remember enough of this stuff and use it infrequently ! Thank you so much !
 

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