Although I ought to know this kind of stuff off the top of my head, I don't. In fact, I seem to forget how to do things immediately after I've done them… What were we talking about? Oh yes, Linear Differential Equations (LDE). Much of engineering and a lot of the 'harder' sciences are about describing the physical world and then using the descriptions to predict something else. Many problems need to describe systems that vary, in a predictable way.

Way back in the 17th century Newton and Leibnitz developed a concise mathematical way of dealing with these situations which has become modern day Calculus. Setting up a mathematical description often results in equations that contain some elements of this sytem and are referred to as differential equations. Certain forms of these are the eponymous LDEs.

So, this is just a simple reminder of what LDEs are, and one way to solve them.

Arguably the best equation in mathematics - well from an engineer's point of view anyway

$$e^{i\theta}=cos{\theta} + i sin{\theta} $$

Apparently this all comes from rotating a unit line in the complex plane - you can Google it if you want to know. There is a special case of this where if you make \(\theta\) equal to \(\pi\) and noting that \(sin \pi = 1\) and \(cos \pi = -1\) it is easy to get Euler's identity:

$$ e^{i\pi} +1 = 0 $$

This is a fascinating equation because it brings together 5 of the most fundamental numbers in maths: \(0, 1, i, e\) and \(\pi\)

Anyway, this post looks at the Euler equation(s) and uses them to do stuff.