Ran into this differential equation, got stuck

[Galorian]

Trying to make a calculation and I ran into the following diff equation:

r''=a+b/r^2

And I can't seem to remember how one would go about solving it.

b = 1355041.84
r(0) = 1400
r(t) = 239.6
r'(o) = 0
r'(t) = 75.2
a = ?

I'm specifically looking to find the value of the constant a. If someone could drop off a quick solution that would be great!

(Note: this is part of an attempt to calculate the maximum acceleration of a starship from a video game cutscene as it approaches a small(is) object with a powerful gravity field)

 

[HallsofIvy]

What do you mean by "r'(t)= 75.2"? What value of "t"? "a" is just a parameter in the problem that has to be determined by additional information. Perhaps "r'(t)= 75.2" is that "additional information" but I don't understand what it means.

 

[Galorian]

What do you mean by "r'(t)= 75.2"? What value of "t"? "a" is just a parameter in the problem that has to be determined by additional information. Perhaps "r'(t)= 75.2" is that "additional information" but I don't understand what it means.

I don't know how long it took the ship to accelerate since the scene skips to the slingshot.

All I know is the distance traveled (from a starting distance of 1400km to a distance of slightly over 239.6km), the velocity of the ship as it reached the closer radius, the fact that the ship wasn't moving at t=0 and the fact that the gravity acceleration of the miniature sun it was slingshotting around was about 1355041.84/r^2.

That should be enough conditions to calculate a final answer, I just can't recall how to solve that kind of a differential equation.

[EDIT] Note that I'm not calculating the gravity slingshot maneuver that was preformed right after that approach, only the approach itself.

 

[Galorian]

Nevermind, I just realized that I could calculate the whole thing via gravitational potential energy and hilariously enough the figure came out as higher than the speed of the ship itself...

Thanks anyway! ^_^