While thinking about the motion of a particle being attracted to another massive particle and considering that the acceleration of that particle is not constant, I found myself trying to solve this equation $\frac{d^2f(t)}{dt^2}=\frac{1}{f(t)^2}$
This is how I got there:
$m_1$ is the mass of object 1 and $m_2$ is the mass of object 2 which we will consider to be much larger than $m_1$.
Newton's law gives us: $F=ma$
$F=G\frac{m_1m_2}{r(t)^2}$ Let r be the distance between the two bodies.
We assume here that the movement of the particle with mass $m_2$ with respect to a stationary point is negligible.
$m_1\frac{dr(t)^2}{dt^2}=G\frac{m_1m_2}{r(t)^2}$
$\frac{dr(t)^2}{dt^2}=G\frac{m_2}{r(t)^2}$
I believe that all of this is a modified version of the two-body problem. Hopefully, the solution will be slightly simpler.
This all leads us to the original question of how to solve:
$\frac{d^2f(t)}{dt^2}=\frac{1}{f(t)^2}$