Mass-spring system; Hill equation

1. Dec 8, 2008

Drokz

When trying to solve a problem I arrive at the following equation of motion / Hill equation:

$$\frac{d^{2}y}{dx^2} + \frac{4 k_0}{m w^2} cos(2x)y = 0$$

There exists a value x_0 such that for all x>x_0 the motion is stable.

I actually don't know what is meant by this 'stability'. Can someone help, please?

Thanks, Drokz

2. Dec 8, 2008

marcusl

I've usually seen Hill's equation with omega^2, not omega^(-2). Is this a typo?

3. Dec 9, 2008

Drokz

I don't think it is a typo. Omega is just a constant here, I think.

4. Dec 9, 2008

marcusl

Ok. Solutions are products of exponential terms and periodic functions, so your stability condition is needed to keep the exponential parts of the solutions bounded. Do you have access to a reference on Mathieu functions like Whittaker and Watson, or Erdelyi's Higher Transcendental Functions? The latter one has a clear discussion of the stable and unstable solution regions (p. 101 vol. 3).