Good Tips to Solve Non-Linear ODE

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Discussion Overview

The discussion revolves around finding a non-trivial non-linear ordinary differential equation (ODE) to demonstrate a solver. Participants explore potential equations that are significant and challenging to solve, focusing on both analytical and numerical methods.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant seeks suggestions for a non-linear ODE that is significant and non-trivial to solve.
  • Another participant proposes the Riccati equation, noting its lack of a general analytical solution and its importance.
  • A participant clarifies their focus on numerical solutions rather than analytical ones, suggesting that analytical methods for such equations may already exist.
  • There is a recognition that while the Riccati equation has no known general analytical solution, numerous numerical methods may be applicable.

Areas of Agreement / Disagreement

Participants do not reach a consensus on a specific equation or method, and multiple competing views regarding the existence of solutions and methods remain present.

Contextual Notes

Limitations include the ambiguity around the significance of the proposed equations and the reliance on existing numerical methods without detailed exploration of their effectiveness.

JoPe
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Hi!
I am trying to demonstrate a solver for non-linear ODE an wonder if anyone has got a tip on one which is non-trivial to solve, and has some significance to some people so that maybe someone will read my report =)

If you got a good tip, thank you very much!
 
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How about this Riccati equation:

[tex]y^{\prime} + y^2 + \alpha(x) = 0[/tex]

(where alpha is an arbitrary function of x, and y = y(x) as well). This has no general solution (as far as I know) -- and it is very important. If you can provide an analytic solution to this, then fame and fortune is yours. ;-)
 
Thank you, i will look into that one. Although it is not an analytical solution i am working with, but a numerical. If i am not mistaking the analytical method for solving them are already known?
 
Sorry, I misunderstood -- the equation I quoted has no known general solution (analytical), but I suspect there are many numerical methods already associated with it. If you apply your method, you should then do a survey on the web of other numerical techniques applied this class of equations, and then compare results.
 

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