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Proving an Autonomous First Order ODE is Bounded

  1. Sep 23, 2009 #1
    1. The problem statement, all variables and given/known data
    For the following auto. first order ode: x' = x^2 - y -1 , y' = x + x*y, show that each integral curve begins inside the unit circle remains there for all future time.

    2. Relevant equations
    Okay, i think what needs to be shown... define a new equation r^2 = x^2 + y^2. Show that if dr/dt = 0, then the radius of the solution is constant, and therefore the ode is bounded. I just do not know how to solve that ode.

    3. The attempt at a solution
    Sorry, this is probably a very simple question and I apologize for the sloppy notation. I first let dy/dx = [(x)(y+1)]/[x^2 - (y+1)]. Then... dx/dy = [x]/[(y+1)] - [1]/[x] . I'm completely lost on how to solve this.
  2. jcsd
  3. Sep 24, 2009 #2


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    Welcome to PF!

    Hi RJq36251! Welcome to PF! :smile:

    (try using the X2 tag just above the Reply box :wink:)
    Not following that :redface:

    you need to show that, if r = 1, then dr/dt < 0 (or if dr/dt = 0, then … ) :smile:
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