Differential Equation @ 2nd Order How to solve this? $$\dfrac{d^2x}{dt^2}=-kx^2$$

I need it in physics problem.https://physics.stackexchange.com/questions/60202/non-shm-oscillatory-motion/     Though solved without it but still it would be better to have this.
The differential equation is not easy to solve. The usual trick is to let $v=\frac{dx}{dt}$. Multiply both sides by $2\frac{dx}{dt}$. 

On the left-hand side we have the derivative of $v^2$. On the right, we have the derivative of $-\frac{2k}{3}x^3$. So $v^2=-\frac{2k}{3}x^3+C$. We end up with a separable equation. Unfortunately, unless we are very lucky and $C=0$, we end up with something that cannot be integrated in elementary terms.
I think this might be part of the solution

let x(t) be a Taylor series

the co-efficients of the taylor series contain x(0), x'(0), x''(0) ... etc 
I think these can be derived from d2xdt2=−kx2 and some initial conditions for x(0) = a, and x'(0) = b

http://i.stack.imgur.com/nRi4k.jpg

The final recurrence relation can be used to compute the x(0), x'(0), x''(0) co-efficients which can then be used to construct the taylor series for x(t)

It may not be nice but I believe it will work ...