Solving (r³)r''' - Gm r' = 0 for orbital mechanics

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The equation I'm trying to solve is (r^3) * r ''' - G*m*r' = 0

the independent variable is t.

I'm not far along in DE's yet to know how to go about this but I was bored and trying to derive something and came across it. Thanks for the help
 

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I think Laplace transforms might work. The Laplace transform of (r^3) * r ''' could probably found using integration by parts. Another option would be series solutions. Perhaps there are also other way to solve it. Hmmm...It looks like you could also use order reduction.
 
A lot of simple looking differential equations are either completely intractable or at least not amenable to elementary methods. Your equation is a good example. Maple gives a solution for it in terms of non-elementary Airy wave functions.