My Maple 9 gave me 3 different solutions (obviously they are connected to each other) for the original equation. Here they are. I think, everything is clear. I beleive, the integrals can be evaluated under some conditions for K and c, e.g.: K=3, c=2, or K=c^2 or K=3*c (all of them are just the examples, not the real conditions leading to explicit solutions for r(t) or t(r)).
Best of luck,