Solving non-linear second order ODE

[bluestorm]

Homework Statement

It will be great if someone could show me some options:
I need to prove the following:

A particle of mass m is attracted toward a fixed point O (the origin) with a force proportional
to its instantaneous distance from O raised to a positive integer power, i.e., according to the force
law F (x) = −k(x^n) . (Note that n must be odd in order for F (x) to be an attractive force!) Suppose
that, at time t = 0, the particle is released from rest at x = x0 . Show that the time T for the particle to reach O is given by the formula:
T = C / ( (xo)^(n-1)/2 )

Homework Equations

Determine explicitly the coefficient C in terms of (i) the Beta function and (ii) Gamma functions

The Attempt at a Solution

i tried solving it manually using definite integrals and arrived at
C = (m(n+1)/(2*K))^-1/2

but this looks nothing like the gamma or beta function
also, I was unable to find a normal method to solve the 2nd order non linear ODE:
x'' = (-k/m)x^n

because of the x^n... all common methods are give for just x or with some x'
Also tried series expansion solution to the ODE... but the t was in several pieces and cannot put it together in the required form

Also tried using Emden-Fowler equation but arrived to a very similar value of C without any relation to the beta or gamma

Any help will be greatly appreciated

 

[gabbagabbahey]

This is fairly straightforward by use of the chain rule:

\ddot{x}=\frac{d}{dt}\dot{x}=\frac{dx}{dt}\frac{d}{dx}\dot{x}=\frac{1}{2}\frac{d}{dx}\dot{x}^2

Which gives you a separable ODE:

\frac{1}{2}\frac{d}{dx}\dot{x}^2=-\frac{k}{m}x^n