Proving an Autonomous First Order ODE is Bounded

  • Thread starter RJq36251
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Homework Statement


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.


Homework 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.


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.
 

Answers and Replies

  • #2
tiny-tim
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Welcome to PF!

Hi RJq36251! Welcome to PF! :smile:

(try using the X2 tag just above the Reply box :wink:)
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.

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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