Q:

Bounds of ordinary differential equation

I need to show that the solution for the following equation $$\ddot{x}=-\log x-1$$ is bounded for every initial condition.
I started by converting to the system $\cases {\dot{x} = y \\ \dot{y}=-\log x - 1}$ and I reached the following expression for phase lines $(x,\pm \sqrt{2(C-x\log x)})$, where $C$ is constant. Clearly the derivative of $x$ is bounded, but I don't know if it is related. Am I in the right direction? Is there something better I can try?

A:

As with any conservative system of the form $\ddot x=-f'(x)$ one gets the first integral
$$
E=\frac12\dot x^2+f(x)
$$
by multiplying with $\dot x$ and integrating. 
All solutions are restricted to the set $f^{-1}((-\infty,E])$ where the constant $E$ is determined by the initial conditions. Now if the graph of $f$ has the form of  a trough, all solutions remain in bounded regions.

