Given the problem:
Find the time solutions of
$x \ddot{x} - \dot{x}^2 = 0$, $x > 1$
which satisfy $x(0) = 1$. Decide which solutions are asymptotically stable.
OK. Finding the solution is quite simple, and it yields:
$x = Ce^{Dt}$
With the initial condition, we get:
$x = e^{Dt}$
What I am a little bit unsure about is the notion of asymptotic stability. The book presents the theory behding this, but has no proper examples. What I get is that if we start off with two solutions for $t_0$ close to one another, then we have asymptotic stability if the absolute value of the difference between the two solutions approach zero as $t$ tends to infinity. In the above example, my intution tells me that this occurs only when $D<0$. If $D>0$, then we have exponential growth, and two solutions will then move further away from each other. If $D=0$, then a slight disturbance which causes $D$ to become positive, will again ruin the stability. However, when $D<0$, $e^{Dt}$ will approach zero as $t$ tends to inifinity no matter what value we choose for $D$. Hence the absolute value of the difference between two solutions will also approach $0$.
I would truly appreciate it if anyone can confirm/disconfirm that my reasoning above is correct!