Second order differential equation

[elevenb]

Homework Statement

http://books.google.co.uk/books?id=...ilibrium length&pg=PA136#v=onepage&q&f=false

Homework Equations

All in the link.

The Attempt at a Solution

The solution is there, its to do with part (c), I would really appreciate it if someone would explain how they make the jumps - especially to the second last equation

 

[AlephZero]

They used the Binomial theorem.
$$\frac{1}{(d + \Delta d)^2} =\frac{1}{d^2}\left(1 + \frac{\Delta d}{d}\right)^{-2}
= \frac{1}{d^2}\left(1 - 2\frac{\Delta d}{d} + \cdots\right)$$

Everything before that looks like straightforward algebra. The last step to find $\omega$ is the standard solution of the differential equation for simple harmonic motion.

 

[elevenb]

Ignore- spotted mistake