Differential equation with $(\frac{dx}{dt})^2$ I have had very little training in differential equations, so this might be a stupid question. Is it okay to solve something like $t = x^2({dx\over dt} )^2$, by doing: $t(dt)^2=t^2(dx)^2 \rightarrow {t^2\over2}dt={x^3\over3}dx$ and then integrating again? 

Not really, no. You could re-write to
\begin{equation}
\frac{dx}{dt} = \frac{\sqrt{t}}{\pm x}
\end{equation}
From which you would try to solve
\begin{equation}
\pm \int x dx = \int \sqrt{t} dt
\end{equation}
