1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

System of nonlinear PDE

  1. Jan 18, 2009 #1
    1. The problem statement, all variables and given/known data
    Hi, i have the following system of equation. In the task is that system have periodic solution and have to be used polar coordinates.

    2. Relevant equations
    x'=1+y-x^2-y^2
    y'=1-x-x^2-y^2

    3. The attempt at a solution
    After transfer to polar system i tried to use the method of variation of parameters, but without success.
     
  2. jcsd
  3. Jan 18, 2009 #2

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    1) Why is this posted in the "preCalculus" section?
    I assume that was a mistake and I will move it to "Calculus and Beyond" homework.

    2) Why is this tltled "PDE"? I see no partial differential equations. I see a system of two ordinary differential equations.

    Changing to polar coordinates looks like a very good idea but I don't know what you mean by "variation of parameters" for a non-linear equation. What equations did you get after changing to polar coordinates?
     
  4. Jan 18, 2009 #3
    Re: System of nonlinear ODE

    I'm sorry for PDE and wrong section
    In the polar coordinates have equations this shape:

    rho' cos(phi)=rho sin(phi)(1+phi')
    rho' sin(phi)=- rho cos(phi)(1+phi')
     
  5. Jan 24, 2009 #4

    epenguin

    User Avatar
    Homework Helper
    Gold Member

    Did you get anywhere yet?
    One thing you might notice straightaway is that any point on the circle x2 + y2 - 1 = 0 you find the equations become those of SHM whose solution is that same circle, so that circle is a solution.
    However it is not SHM in general, for no other points have that property and (0, 0) is not a stationary point.

    Do you know how to analyse such systems qualitatively? This one appears quite complex and surprising.
    Main thing, you have to find the stationary points (i.e. where x' = y' = 0) and analyse the stability of the linear approximation around them.

    Perhaps the d.e. s can be solved too, I don't know yet.
     
    Last edited: Jan 25, 2009
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: System of nonlinear PDE
  1. Nonlinear PDE (Replies: 1)

Loading...