Since Eta said he'd come with a solution to this problem I thought I'd bump it.
In doing so I'll also propose a new attempt at a solution.
I'll start off from the realisation that the problem could be reduced to the form:
a = G*M/r^2.
This can be rewritten:
r^2*dv = G*M*dt
Which can be rewritten as:
(A) dt = (1/GM)*r^2*dv
Now we know (of course) that:
dr/dt = v
Which can be rewritten as:
(B) v*dt = dr.
Now inserting (A) into (B) we gain:
(1/(G*M))*r^2*v*dv = dr
Which can be rewritten as:
(1/(G*M))*v*dv = dr/r^2
Integrating from v = 0 to v = u on the left hand side and from r = A to r = R (R < A) on the right hand side we gain:
(1/(2*G*M))*u^2 = 1/R - 1/A
Which gives us "u":
u = sqrt(2*G*M)*sqrt(1/R - 1/A) = dR/dt
Which yields:
dt = dR/(sqrt(2*G*M)*sqrt(1/R - 1/A) = K*dR*R/sqrt(1 - R/A)
{Where K = 1/sqrt(2*G*M)}
What follows is a rather tedious calculation where I find the antiderivative of the function on the right hand side.
In this I perform a substitution of variables such that R = p^2 (don't ask why
):
I gain:
- sqrt(1 - p^2/A)*(A*p^2 + A^2*p + A^3)
Which is the primitive function expressed in p.
Integrating (C) from t = 0 to t= t and from p = P to p = 0 we now get:
t = 2*K*[- sqrt( 1 - P^2/A)*(A*P^2 + A^2*P + A^3) + A^3]
Where P^2 = A, so t = 2*K*A^3
t = 2*A^3/sqrt(2*G*M);
I think this is right... but then again I thought my other solution was correct as well, so I remain skeptic.