How to prove the boundedness of the solutions of a nonlinear differential equation I have the following differential equation:
$$
\ddot{x} = -\log(x) - 1
$$
and I need to prove that every solution of this equation is a bounded function. From the phase plane portrait, it is obvious that this is true:

How can I construct a formal proof for this?

$x_1=x$, $x_2=\dot{x}$
$\dot{x}_1=x_2$
$\dot{x}_2=-\log(x_1)-1$
we know that to have the solution for the ODE we need $x_1>0$ 
consider the following function
$$V=(x_1\log(x_1)+0.5*x_2^2)$$
the derivative of this function along the system trajectories is
$$\dot{V}=x_2\log(x_1)+x_2+x_2(-\log(x_1)-1)=0$$
Therefore, the solutions are bounded!
