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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 $v(t)$?

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

Chet Miller
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  • Hi Chet, thank you for the answer, but I was looking for an axpression where I dont have to use the last equation in the linked question to find x(t), your answer gives t as a function of x – Alex Dec 24 '23 at 15:32