Solve third-order nonlinear differential equation with certain boundary conditions

Question

A (distinguished theoretical--not "computer savvy") physicist colleague (C. P.) of mine sent me this ("not quite up my alley") question.

I take it that $G$ is the gravitational constant.
The stack exchange pointed me to
NDSolve third order PDE with boundary conditions
but I think C. P. is thinking of exact--not numerical--solutions.
Assuming that $M_0$ is an independent constant, not simply M[0], my Mathematica formulation of the question put by C. P. is
DSolve[{(1/r^2) D[
  r^2 (1 - 2 G M[r]/r) D[Sqrt[M'[r]/(4 Pi  M0 r^2)], r], r] - 
M0^2 Sqrt[M'[r]/(4 Pi  M0 r^2)] == 0, M[0] == 0, M'[0] == 0, M''[0] == 0}, M[r], r]

Execution of the command does not yield any output. (Nor do limited attempts with NDSolve, with $G=1,M_0=1$,{r,0,1},....)
So, should this be the end of the story?

Paul B. Slater · Answer

I passed this question, along with the several accompanying comments, on back to the initial poser--who responded in an email with an attached pdf along with the remarks: "I attach the Mathematica file with graphs.  The way to solve it analytically is via an iteration procedure like the Hartree Fock method.  Yes, at r = 0, there is a singularity."
Although I seemed unable to post the indicated pdf I was sent here, I copied and executed the Mathematica commands. Here is a url displaying the resultant notebook.
ThirdOrderDiffEq.nb

Solve third-order nonlinear differential equation with certain boundary conditions

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