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2nd Order Nonlinear Differential Equation

  1. Sep 7, 2014 #1
    Hi everyone,

    I need some help to solve this differential equation.

    The question states "Use the perturbation or multiple scale method to find the third-order approximate solution for the following system:

    diff(x(t), t, t)+w^2*x(t)*(1+epsilon*x(t)^2) = 0 "

    Currently, I am still reading and studying on this perturbation method.

    Any advise or directions of what I need to do is much appreciated.
    Thank you for your help in advance.


    Attached Files:

  2. jcsd
  3. Sep 7, 2014 #2
    Hi !

    First, solve the non-perturbated ODE.
    Then, add a term with coefficient epsilon (see attachment) :

    Attached Files:

  4. Sep 7, 2014 #3
    Hi JJacquelin

    Thank you very much for your help,
    I will try it and see if I could reach the result in that picture.
    Thank you again for helping me out.
  5. Sep 8, 2014 #4
    Hi JJacquelin,

    Line 7 on your attached picture, I had a little different than yours. I had X^3 instead of X^2.
    I am still working on it. Please let me know.
    Thank you,

    Attached Files:

  6. Sep 9, 2014 #5
    Hi Jhonmm !

    You are right it isn't X^2 but X^3. This changes some terms of my comments, but this doesn't change the method to solve the problem. I suppose that you can continue by yourself.
  7. Sep 9, 2014 #6
    Hi JJacquelin,
    Thank you for the help.
    I will follow this method to the rest.
  8. Sep 9, 2014 #7


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    "Perturbated"??? Isn't the standard "perturbed" simpler and better?
  9. Sep 9, 2014 #8
    I think it is perturbed.
  10. Sep 11, 2014 #9
    You should probably give us some boundary conditions since otherwise you might generate solutions growing without bound.
  11. Sep 11, 2014 #10
    The question didn't give boundary conditions. But I think it should be x(0)=x0 and [itex]\dot{x(0)}[/itex]=[itex]\dot{x}[/itex]0.
  12. Sep 12, 2014 #11
    Maybe take a look at http://en.wikipedia.org/wiki/Poincaré–Lindstedt_method. Or you could expand using 2 time scales instead using for example
    \frac{d^{2}x}{dt^{2}} = \partial_{\tau\tau}x_{0} + \epsilon\left( \partial_{\tau\tau}x_{1} + 2\partial_{\tau T}x_{0} \right)
    Differential equations with oscillating solutions doesn't always behave nice when using the usual perturbative approach.

    Edit: The wiki link actually treats the same equation as you are looking at! Sometimes you're lucky I guess.
  13. Sep 12, 2014 #12
    Thank you for your reply. I will take a look into the wiki link. :smile:
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