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.

 

[tacman]

First of all, congratulations on tackling a challenging problem and finding a solution! Deriving parametric equations can be a tricky task, but with practice and understanding of the concept, it can become easier.

To answer your question about the last two parametric equations, it seems that they were derived using a computer program, as you mentioned. However, there are a few ways you can approach deriving parametric equations for a given problem.

One way is to use a graphical analysis, as you have already done. This involves plotting the given equations on a graph and analyzing the behavior of the curves. By looking at the slopes and points of intersection, you can come up with the parametric equations.

Another method is to use algebraic manipulation. This involves manipulating the given equations to isolate the variables and then using the parametric equations x = f(t) and y = g(t) to express the variables in terms of a third variable, t. This method requires a good understanding of algebra and manipulation techniques.

As for references, there are many textbooks and online resources that cover parametric equations and their derivation. Some popular textbooks include "Calculus: Early Transcendentals" by James Stewart and "Calculus" by Michael Spivak. You can also find helpful videos and tutorials on websites such as Khan Academy and MathIsPower4U.

In summary, deriving parametric equations can be done through graphical analysis or algebraic manipulation. It may take some practice and understanding of the concept, but with determination, you can master this skill. Keep exploring and learning, and you will continue to improve in this area. Best of luck on your mathematical journey!