Q:

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?

A:

What  about assuming a solution (boundary conditions permitting, you did not mention any) such as $$ y = e^{\beta x}$$
Then one gets $$ \beta y \beta^2 y = k y^2 $$ from which the condition $$ \beta = k^{1/3}$$ is derived. 

A:

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.

