# Perturbation Techniques and Theory for Nonlinear Systems

## Homework Statement

Given the equation
$$\ddot{\theta}=\Omega^2\sin{\theta}\cos{\theta}-\frac{g}{R}\sin{\theta}$$
Determine a first-order uniform expansion for small but finite theta.

## Homework Equations

Other than the equation above, none so far as I am aware.

## The Attempt at a Solution

The only thing I could think to do was try to solve this differential equation via the method of undetermined coefficients, which I do not think is right at all. I then planned to expand my solution in a Taylor series about 0. This is from Ali Hasan Nayfeh's Introduction to Perturbation Techniques. My professor gave us a packet of the fourth chapter of the aforementioned text as a basis to solve this and other problems. Nowhere in the text does it give a clear example of what exactly a "first order uniform expansion" is, nor do I even know where to begin. My professor's research interests lie in nonlinear dynamics and chaos, and I fear he is going a little too in depth for my second year physics course. Thank you for any input.

vela
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