\[\dot{\bm{x}}=\left(
\begin{array}{c}
\frac g\omega g\cdot\ln(\omega t+1)+a\\gt+c\\0
\end{array}\right)
\]
\[\bm{x}=\left(
\begin{array}{c} \frac g\omega^2\left((\omega t+1)\cdot\ln(\omega t+1)-\omega t\right)+at+b\\\frac12gt^2+ct+d\\0
\end{array}\right)
\]