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Homework Help: 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

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    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

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    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
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