Asymptotic convergence of ODE solutions to a unique function, regardless of initial conditions I have a non-linear differential equation of the form
$$
\frac{dy}{dx}=F(x) - G(y,x)
$$
where $G$ is of the form
$$
G(y,x) = y^3f_3(x)-y^2f_2(x)
$$
and $f_2,f_3>0$ for all $x$, and $F(x)>0$ for all $x$.
Below I attach results from numerical integration for different initial conditions (the integration is backwards from $x=3$ to $x=0$), indicated by the colored lines.
It seems that regardless of the initial condition, the solutions seem to be asymptotic to a unique function.
Is there a general statement I can use to prove this result? Namely, what conditions do $F,G$ need to satisfy in order for solutions to be asymptotic to a unique function, regardless of the initial conditions?
Furthermore, it seems that a good approximation for this asymptotic behavior is given by the solution to the equation (shown as the black dashed line)
$$
G(y,x)=F(x)
$$
which one gets if he sets $dy/dx=0$ in the above equation (which, of course, is not true). How can one explain this?


If the rate of change of the $f_k$ coefficient functions is small enough, relative to their values, or the $y$-derivative close to the root of the polynomial, then you can set $x$ as constant and apply the rules of an autonomous scalar dynamical system.
Close to the roots (as long as they are simple) the solution either moves to them or away from them exponentially. If far away from the roots, the solution changes fast in vertical direction towards one of the roots, as you can see in your initial segment from $3.0$ to $2.8$. As the coefficients change "adiabatically", the solution will follow that change, lagging in time. This is what you observe moving from $2.8$ to the left, with the solution slightly below the root curve.
