On this day in 1707, Leonhard Euler was born. He was one of the greatest mathematicians ever. And that gives me an opportunity to discuss one of my favorite subjects: differential equations. I love them because they are such an obvious example of creativity in mathematics. Although courses are taught in them, there isn’t a lot to learn. Solving them is a highly intuitive thing, and people who are very good at them are like magicians.
Let’s consider the Cauchy–Euler equation. In its most common form, it looks like this:
What the dy/dx represents is the instantaneous trend line: the change of y relative to x. And the figure that looks like the square of that is the instantaneous trend line of the instantaneous trend line. Our job is to find the equation for y without without the differentials.
So how do we proceed? Well, this is a standard equation and we know exactly what to do. But that wasn’t true hundreds of years ago. And that’s where the magic comes in. I should note, however, that most differential equations have no solutions. Anyway, we proceed by making an assumption that:
Substituting this into the equation above causes all of the x terms to fall out and leaves us with a simple quadratic formula for m:
Of course, this equation is going to give us two roots (although they may be identical). And the roots may be real or imaginary (involving the square root of a negative number). In the second case you get exponential solutions. But the two real roots case provides an answer like this:
What all of this shows is that the process is a whole lot of intuition guided by experience. This is why I say that mathematics is at least as creative as art. People who don’t understand math tend to think it is about applying a bunch of rules. But if that were true, math would be simple. It is actually constrained creativity. And that’s true of any kind of art. Except with mathematics, the constraints are entirely internal. And with that, join me in saying…
Happy birthday Leonhard Euler!