RJq36251
Sep23-09, 09:12 PM
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.
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.