There are two equations arising in the context of a physical problem from a paper I am reading: \begin{align} \dot{\zeta}+\zeta &=1\\ \ddot{x}+\dot{x}&=\frac{\zeta}{x^3} \end{align} in which $x,\zeta,$ are functions of time $t$ and dot superscript is the time derivative operator. All quantities are dimensionless.
Now above equations are simplified by employing some heuristic arguments. First, as $t\to\infty$, $\zeta\to 1$, which I understand by obtaining exact solution of first equation.
The second argument is what I find hard to decipher: It is said that as time $t\to\infty$ (and therefore $\zeta\to 1$) and $x\to\infty$, in the second equation $\ddot{x}$ becomes negligible in comparison to $\dot{x}$; presumably what they mean is $\ddot{x}/\dot{x}\to 0$. The second equation cannot be solved exactly (as far as I know) so how could one arrive at such a conclusion? Any help is much appreciated.
Let $\Omega$ be a really large number and consider $X(T)=x(ΩT)/\sqrt[4]Ω$, so that with the $T$ scale we consider the time axis from afar.
Then $\dot X(T)=Ω\dot x(ΩT)/\sqrt[4]Ω$ and $\ddot X(T)=Ω^2\ddot x(ΩT)/\sqrt[4]Ω$ so that the differential equation for $X$ is $$ \frac1Ω\ddot X(T)+\dot X(T)=\frac{ζ(ΩT)}{X(T)^3} $$ In this formulation the second derivative has indeed a small coefficient and can be removed for a first approximation. With $ζ(ΩT)\approx 1$ one gets $$ X(T)=\sqrt[4]{X(T_0)^4+4(T-T_0)},\\ x(t)=\sqrt[4]{x(ΩT_0)^4+4(t-ΩT_0)},\\ $$