x(t) = [-9A/2]^(1/3) * t^(2/3)
Well, is this the only solution?
That is a very good question Wasi.
At first I was thinking exponential but by manipulating I got:
[dx/dt]^2 = -2A * d/dt(1/x)
and then guessed at a power function: x = C*t^m
So x^2 = C^2*t^(2m) and x'' = Cm(m-1) * t^(m-2)
That gave me 2m+m-2 = 0 and m = 2/3. Similar for C by equating co-efficient to A.
It reminded me of Newton's Law of Gravitation:
acceleration is prop. the inverse square of position.