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.