Finding speed as a function of time from acceleration as a nonlinear function of position

Question

For general function $x'' = G(x)$ with $x(t=0) = x_0$ and $x'(t=0) = v_0$, what could the general form of $x'(t)$ be?

I was able to do a derivation similar to:

Convert acceleration as a function of position to acceleration as a function of time?

But it only gives velocity as a function of position. Is it possible to get a simpler expression of $v(t)$ without using the last integral in that post to find $x(t)$ and subtituting it to find $v(t)$?

Answer 1

$$v(x)=\frac{dx}{dt}$$ $$dt=\frac{dx}{v(x)}$$ $$t=\int{\frac{dx}{v(x)}}$$