Solving an nonlinear ODE $$y'y'' = ky^2$$
I need a closed form expresion, if not atleast an almost closed form expression such as an inverse of an integral characterization. What could be the properties of its solutions?

Multiply both sides by $y'$ and integrate to get $y'^3=ky^3+C$. This integrates again to give a very ugly hypergeometric function (according to Wolfram alpha). So just hope your boundary conditions make C=0 to give An aedonist's solution.
