$x'=x^2$ unstable solutions when $x(0)\geq 0$ but asymptotically stable when $x(0)\leq0$ How do I show that the differential equation $x'=x^2$ has unstable solutions when $x(0)\geq 0$ but asymptotically stable solutions when $x(0)\leq0$?
Usually, I look at the eigenvalues of the matrix to determine stability of solutions, but since there is no matrix here, how do I approach this?
Edit: the solutions are $x(t)=\frac{1}{c-t}$, but what can I do with this?

suppose $x(0) = k.$  then the solution is $$x = \frac 1{t + 1/k }, -1/k < t < \infty$$ the graph is monotone increasing on the domain and $\lim_{t \to \infty} x(t) = 0.$ 
this solution approaches zero but not expentailly; it approaches zero like $\frac1t.$ i don't this the solution is asymptotically stable; just stable. 
