Diffy Q's

My cousin, who's now an engineer, hated school and even Calculus until he finally took a class he could use: Differential Equations (affectionately known as "diffy q's").

Your algebra, geometry and precalculus teachers will tell you the reason to learn all the stuff in their classes is that you'll need it in calculus. But what's calculus for?

In the 1600s Descartes put algebra and geometry together for the first time, allowing others like Fermat to make progress on finding slopes of curved functions and areas under curves. Newton and Leibniz systematized the process and it became Calculus with a capital C.

But it all would have remained abstract, "pure" math if not for the fact that scientists were able to use the new math notation (Leibniz's, anyway) for modeling situations they found in nature. Situations where you didn't know the value for something (heat, for example) but you knew how fast it changed were all of a sudden able to be modeled using differential equations like dy/dt = ky.

Of course Newton made the most notable use of his new invention to solve the differential equations he encountered in his astronomical studies. His solutions proved Kepler's laws to be correct and that his own famous laws applied to objects on earth and in the heavens.

Textbooks are written every day about solving differential equations using algebra and integration. Instead of the change in y you get a formula for y itself. Take the differential equation

dy/dx = y - x

We're looking for a function y whose derivative is the original function y minus x. In school you jump through all kinds of hoops to find the solution for y is

y = ce^x + x + 1

Take the derivative of that function and sure enough, it's the function minus x. Well, those differential equations have all been reduced to algebraic formulas thousands of times, and more with every school year. Furthermore, most of the differential equations that are encountered in the real world have no explicit solution with x's and y's. This situation was already well known in the 1700s, leading Euler to develop "numerical methods" of approximating solutions using "Euler's Method."

If you know one value (an initial value, say) of y but otherwise you only know how it changes, you can start at that value and just take steps using the direction given by the differential equation. If the steps are small enough, you can get a good approximate graph of the function y. Euler's Method has been improved over the years, and now it's even made into highly instructive Java applets. Every so often somebody codes these methods into a new free differential equation solver software to help, like Calcode or Maxima.

So we do Algebra and Geometry to do Calculus, but we do Calculus in order to solve Differential Equations. The faster we get to Diffy Q's the better. I introduce them in the third meeting of the Math Through Technology Program, starting September 6th.