How Are Parametric Equations Derived in Complex Mathematical Problems?

[GCT]

I've recently attempted the following problem,

http://domino.research.ibm.com/Comm/wwwr_ponder.nsf/challenges/June2001.html

with the following solution

http://domino.research.ibm.com/Comm/wwwr_ponder.nsf/solutions/June2001.html


I've managed to form the meaningful derivatives (as they have listed on the solutions page) through a graphical analysis, but have no idea how they came up with the last two parametric equations. I wasn't even familiar with the term, and am getting myself up to date-any useful references...texts, internet site, specifically for this subject would be appreciated.

I would appreciate any comments regarding how they have derived the last two parametric equations on the solutions page, though it seems that the equation might have been computer derived.

 

[Manchot]

It appears that they solved the system of DE's that they derived and plugged in the initial values. I only spent a few minutes on it, but unless there's some "special trick" they used to solve it, I'd imagine that they did in fact use a computer program.