Stability of non-linear, non-autonomous ODE Consider the ODE  
$y'(t)=S(K(t)y(t)+b(t))$
where $K(t)$ is a matrix, $y(t), b(t)$ are vectors and $S$ applies a function $\mathbb{R}\to\mathbb{R}$ componentwise. Can the stability of this ODE be assured by chosing $K$ such that the Jacobian of the right-hand side  only has eigenvalues with negative real part at each instant ?
I read a paper where this is claimed, but I cannot find a version of this criterion for non-autonomous, non-linear systems. Also why can we talk about the whole ODE being stable if generally speaking for non-linear ODEs we can only speak about stability for specific solutions ?
 A: In differential equations there are different notions of stability:
First we make the distinction between global and local stability.  Local stability, in general is easier to show.  There are several approaches to doing so.
The Hartman-Grobman Theorem is generally the tool of first resort for showing local asymptotic stability of autonomous non-linear systems(i.e. if I start sufficiently close to an equilibrium value, then over time I will approach arbitrarily close to said point) in non-linear ODEs.  It is useful, in part, because for most such equations it is at best non-trivial to find an explicit solution to the system.
The condition you mention is a consequence of the requirement of this theorem that the equilibrium be hyperbolic, that is that the real part of the eigenvalues of the Jacobian matrix (that is the linearization) evaluated at this point are non-zero.
In the event of a non-hyperbolic equilibrium center manifold theory is one approach to analyzing model behavior 
Another notion of local stability is Lyapunov Stability, which is the idea that if I start sufficiently close to an equilibrium then I stay within a bounded distance of that equilibrium point for all time.
This is generally shown to be true by trying to find a Lyapunov function (a task requiring luck and experience)
One can also show local asymptotic stability in this way if the Lyapunov function is strict ($\dot{V} < 0$ for $x$ different from equilibrium $x^* $), or, sometimes, using LaSalle’s Invariance principle 
Global asymptotic stability is the notion that no matter where I start in relation to an equilibrium solutions will converge arbitrarily close to said equilibrium.  This is usually quite difficult to prove in nonlinear systems, and there are a variety of approaches to doing so, e.g. some extensions of arguments for local stability using Lyapunov Functions.
I also think you might have a bit of confusion about terminology.  Equilibrium, i.e. critical points of some vector field (i.e.
$x’(x^*) = 0$) are constant solutions of the ODE and may be individually stable.  It is entirely possible for multiple equilibrium to be simultaneously stable, but not, generally, for all of them to be.  Without reading the specific paper I would assume that by this they mean that there is an equilibrium where all compartments of $y$ are nonzero is stable, or they are talking about a scalar ODE.  While linear ODE have one equilibrium (at 0) nonlinear ODE may have many.
Non-autonomous systems are much trickier to analyze qualitatively, and are often simply solved numerically.  That said one approach to doing analysis is via the Lyapunov theory I mentioned earlier.
