I am wondering if anyone has any advice to try to find an explicit solution (or some approximation) for the following non-linear differential equation system:
$\frac{dx}{dt} = A - B y(t)$
$\frac{dy}{dt} = C x(t) - \frac{D}{y(t)}$
Is there any method to try to arrange these equations to find an explicit solution? Or define some constraints in the solver to find a solution for some restricted domain? I know the constants are real and positive and the region of interest is x and y > 0 (same for initial conditions).
By using some numerical values, I found the solutions look like some kind of spiral. I would like to have some way to characterize these curves (like the position of the center, radius, etc...) and how A,B,C and D, somehow modify the shape of the solutions. Any help, hint or advice is greatly appreciated!
[EDITED]:
By using MATLAB I have found the following solution:
The solutions seem to have a spiral shape, but they diverge from the center at some point. Is there any way to find what's the curve that separates the solutions that move to the "left" from the ones that go back to the "right"?
Thanks!
One hint from the equations is that the spiral can't be correct for solutions in the positive quadrant; since the derivatives of $x$ and $y$ are positive, they both increase. That prevents any type of circulating behavior. If you can give some context for the problem,that might help get better answers.