The k=+1 solution should look like the following but I cannot get it into the same plot as the k=-1,0 solutions.

@Kitonum This sort of works but I am trying to plot the 3 k=-1,0,1 solutions which are:

and

I know that one of them will expand and then collapse back to 0, and the other two will expand forever. I used the following however one of the curves cuts short and I believe it is the one that should collapse.

with(plots);
implicitplot({S = t^(2/3), sqrt(S)*sqrt(1+S)-arcsinh(sqrt(S)) = t, arcsin(sqrt(S))-sqrt(S)*sqrt(1-S) = t}, t = 0 .. 3, S = 0 .. 3);