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16 January 2004

A New Year, A New Column

by George E. Hrabovsky, President, MAST

Hello, Welcome Back

A lot has happened in the last year; some good, some bad.  One thing that happened was a period of time where we had no Internet access (nor even a phone). This has given me a period of reflection.  What is the purpose of the Mathematics Corner?

I have decided to make this less of a course of lectures on mathematics, and bring it more into line with the other write-ups in the E-Bulletin.  To that end, and to make the mathematics more useful, I will give step-by-step instructions on how to do specific things; whether they be problem-solving, theorem proving, or formulating a hypothesis.

What to Include

My intent is not to cover mathematics like a course or an encyclopedia.  My goal is to write about the mathematics that interests me.  This includes differential geometry, topology, differential equations, and abstract algebra.  In order to write about these things, I will need to cover background material.  Let's say that I want to write an article about the geometric properties of curves.  To do this I need to make sure you understand how differential calculus relates to vectors.  This will involve a series of articles (and I will tell you this in advance).

A First Goal

I will set my first goal as learning to formulate and solve differential equations.  We have a lot of the machinery to do this already.  Also the mathematics of differential equations touches upon each area of mathematics that interests me.

A First Question

It is always a good idea to start an investigation with a question.  "What is a differential equation?"

Let's take a stab at it.  I would start with the answer, "A differential equation is an equation containing one or more derivatives."

Solving such an equation is tricky.  For example

y/x = x^2

is a differential equation.  How do we solve for y?  We can certainly manipulate the equation to get to,

y = x^2x .

This is a simple example of what is called the separation of variables.  This is nothing more than making sure all variables have thier own side of the equation.  How do we proceed from this point?

It turns out that this is directly related to integration via a principle called the antiderivative of a function.  We can write,

F ' (x) = F(x)/x = f(x)

where F(x) is the antiderivative of the function f(x).  So we ask ourselves what function f(x) is the derivative of.  For example,

f(x) = 2x ⟹ F(x) = x^2

f(x) = sin x ⟹ F(x) = -cos x

f(x) = x ⟹ F(x) = x^2/2

and so on.

We can define the indefinite integral as,

∫f(x) x = F(x) + C

where C is called a constant of integration and can be determined often by standard algebraic solution methods.

Mathematics Challenge

Solve the differential equation given above by indefinite integration.

Created by Mathematica  (January 14, 2004)