Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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.

    John
     

    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

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    "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:
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook