Q:

Approximating the time it takes for a particle with a potential $-Ax^4$ to approach the origin

Here's the problem I'm solving:

A particle of mass $m$ can only move along the $x$-axis and is subject to an interaction described by the potential energy function $U\left(x\right) = -Ax^4$, where $A > 0$ is a constant.  The particle's energy is $E = 0$ and its position at $t = 0$ is $x_0 > 0$.
Assume that $v_0 < 0$ and estimate (analytically) how long it would take for this particle to approach $x = 0$.

I solved the equation of motion and got $$t = \sqrt{\frac{m}{2A}}\left(\frac{1}{x} - \frac{1}{x_0}\right).$$
The time it takes for the particle to reach $x = 0$ is $t = \infty$, but it's clear to me that this isn't what's being asked for.  I don't know how to approach this problem and I'd like some guidance. 

A:

You're probably not supposed to actually solve it, just estimate it using some simple approximations.  For example, you could figure out the initial velocity $v$ and then take $x_0/v$ as a rough timescale.