Find an equation for position vs time given potential vs position?

[razgriz129]

Homework Statement

If I was given the potential energy equation
$$U(x)=4x^2$$
How would I find the equation for position vs time? I know the graph for position vs time for this equation is a sin curve because the book tells me so.

$v = 0$ and $x = 0$ when $t = 0$

Homework Equations

$$\frac{d}{dx} U(x) = -F(x)$$

The Attempt at a Solution

Taking the derivative of the potential energy equation would yield an equation of force. $$-F = 8x$$

Using Newton's second law, we get: $$ma = -8x$$

Replace acceleration with velocity and we get: $$m \frac{dv}{dt} = -8x$$

I know I would integrate here twice somehow to find position in terms of time, but I'm not entirely sure how. In addition, I don't see how this position equation would have a sin function, as the book shows it should. I was hoping to find a way to solve this or learn a different approach here.

Thanks in advance.

 

[voko]

Could you use conservation of energy here?