Good Tips to Solve Non-Linear ODE

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SUMMARY

The discussion focuses on solving the Riccati equation, specifically the non-linear ordinary differential equation (ODE) represented as y' + y² + α(x) = 0, where α is an arbitrary function of x. It is established that this equation lacks a general analytical solution, making it significant for researchers and practitioners in the field. The conversation emphasizes the importance of numerical methods for solving such equations and suggests conducting a survey of existing numerical techniques to compare results effectively.

PREREQUISITES
  • Understanding of non-linear ordinary differential equations (ODEs)
  • Familiarity with the Riccati equation and its properties
  • Knowledge of numerical methods for solving differential equations
  • Experience with analytical versus numerical solution techniques
NEXT STEPS
  • Research numerical methods for solving the Riccati equation
  • Explore existing literature on non-linear ODEs and their applications
  • Learn about specific numerical techniques such as Runge-Kutta methods
  • Conduct a survey of various numerical solutions applied to non-linear ODEs
USEFUL FOR

Mathematicians, engineers, and researchers working with differential equations, particularly those focusing on non-linear ODEs and numerical analysis techniques.

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:

y^{\prime} + y^2 + \alpha(x) = 0

(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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