Proving an Autonomous First Order ODE is Bounded

[RJq36251]

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.

 

[tiny-tim]

Welcome to PF!

Hi RJq36251! Welcome to PF!

(try using the X² tag just above the Reply box )

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 …

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