I have the following problem. I need to get the equation for motion of a particle. So solve for $r(t)$ Newtonian physics but I'm running into a bit of a problem.
The parts I'm reasonably sure about led me to:
$$\frac{d^2r(t)}{ dt^2} =\frac{ G \times M}{ d(t)}$$ for $G,M$ constant
$r$ is position, $t$ is time, $d$ is distance to (in this case fixed at $(0,0,0)$ object (so maybe $d(t) = r(t)$ ?)
Solved it on wolfram alpha, and got an interesting result
$$r(t)=\int_1^t\left(\int_1^\zeta\frac{cG}{d(\xi)^2}\text{d}\xi\right)\text{d}\zeta+k_2t+k_1$$
I understand that there is some variable substitution going on there but how do I work with this? If I want to get values for specific $t$'s or $r$'s do I just put distance at that time on $\xi$? Ignoring the constants from derivatives can I get a one solution?
In reduced units,
$$\ddot r=\frac1r$$
or
$$2\dot r\ddot r=\frac{2\dot r}{r}.$$
Hence by integration
$$\dot r^2-\dot r_0^2=2\log\frac{r}{r_0}.$$
This is separable, but integration does not seem easy.