Introduction: Differential Equations

Differential Equations have an absolute massive range of uses and are one of the most useful applications from maths to the real world, they have massive influences in: quantum mechanics, electromagnetic equations, biology, economics and an absolute ton more in all of physics; read more about notable differential equations. First I need to explain what a differential equation exactly is. A differential equation is any equation that involves a derivative of any degree, examples of differential equations are:

Differential equations can have different orders, when only the first derivative is present it is a first order differential equation. When there is a second derivative is involved it is said to be a differential equation of the second order. Differential equations of higher orders follow the same pattern for what order it is.

Differential equations can be either linear or non-linear. A differential equation is said to be linear if the highest order derivative of the dependent variable y can be expressed as a linear function of y and the
lower order derivatives. Hence a second order differential equation is said to be linear it must be possible to express it in the form:

To solve a differential equation you wish to find y in terms of x. Some of these are very easy to solve and can be solved by algebra and calculus methods (analytical methods), and you may notice that the first of these examples is very easy to solve and can be solved with calculus. All we need to do is integrate and we will get the answer for all values of y as y = x³ + c. But often it is not that easy to solve and you can not easily express y in terms of x, when this is the case numerical methods are very, very useful.

A numerical method is one that will find approximate solutions to the equation at a point you wish to find. There are a lot of numerical methods to solving differential equations, some more effective that others. I will introduce just the one of these in this post.

Euler's Method for Finding Solutions to Differential Equations
This method was created by the mathematician who seems to have influenced all areas of maths in some way or another, Leonard Euler. If we have some unknown function, but we have what its derivative equals and where one point is on the unknown function then we can find approximate y-values for a point further along the curve.

This method is most effective for small values of h. It can of course be repeated to give a more accurate value for y. If we know a point on some function is (3,5) and we want to find the point 3.4, rather than using a step size of 0.4, we could use one 0.1 and just repeated the algorithm 4 times. This will be more accurate because it will map the graph and it's gradient at each step of 0.1, rather than assuming the graph will have the same gradient the whole time.

This is just an introduction to what differential equations are and one relatively simple method of solving them. If you liked this please like us on Facebook.

Basic Introduction to Topology

Topology is the study of properties that are preserved within objects through twisting, stretching and deformations of that object. However you are not allowed to tear the object, this is key.


In more formal terms, topology is the study of shapes without reference to distance. Or in yet more formal terms it is the study of continuous functions. The study of continuous functions requires the knowledge of range and domain of a function, it studies functions with domains and ranges that make the function continuous. But if you knew that, it is unlikely you would be reading this blog post for absolute beginners as to what topology is!


Also a slight detour from the topic, a continuous function is one where a small change in the input results in a small change in the output too, if it does not follow this then it is said to be discontinuous. Also, just to note, if the inverse function of a continuous function is also continuous it is said to be bicontinuous.


A good way to picture the rules of topology is to picture plasticine, if you have a shape that can be twisted, stretched or deformed in anyway without being torn and put together into a new form then your original shape is topologically equivalent to the new shape you would make.


The reason that topology is useful is that it examines the properties of a shape or region and investigates the properties that are independent of geometric properties the shape or region has. This is the great advantage that topology utilises is that it does not rely on angles or distances, the only thing you need to know is the type of "continuous information" of the object.


Two objects with the same topological properties are said to be homeomorphic. The most well known example of this is that a torus and coffee cup are homeomorphic, they are (topologically speaking) the same shape. As a lot of objects are homeomorphic, it means that if we have a theorem that applies to one object in topology then it also applies to every other object that is homeomorphic to it.

An animation depicting how you
create a coffee cup from a torus.

The point of topology is not to be confusing or just strange, like all Maths it serves some purpose. Be that to merely solve more problems in Maths with a greater ease or to answer problems in the "real world", there is always a purpose to a Mathematical field. And of course, topology is no different. Topology was created to investigate the relationships between objects, and that is exactly what topology does! In fact the origins of topology stemmed from the famed Leonhard Euler when he tried to solve the problem Seven Bridges of Konigsberg, however it truly took off when George Cantor and Henri Poincaré. Poincaré (the famous Mathematician) also famously said "mathematics is not the study of the objects themselves, but the relations between them".


Previously unanswerable questions are now possible, because of the aid of topology. Some that you may wonder why anyone ever wanted to know and others that are pretty useful and interesting. This list should show you what I mean by that.

1. If a coffee is stirred gently then is there always a drop of coffee that finishes in the same place as it started.
2. It is not possible for the velocity of the wind to be non-zero at every point on Earth. This means that at some point at Earth, this means that at any time at all there is no wind whatsoever.
3. There are always two points on the equator that are 180° apart in longitude that have the same temperature.
4. A tennis ball can never be combed in such a way that a cowlick does not appear.
5. It is always possible to cut two irregular pancakes on a plate in half using just one slice.

And that is it for a basic introduction to topology, I will try to get more posts done on topology in the future, so please stay tuned for that! Be sure to comment any questions, tell your friends about us and like us on Facebook!