**Real life use of Differential Equations**

Differential equations have a remarkable ability to predict the world around us. They are used in a wide variety of disciplines, from biology, economics, physics, chemistry and engineering. They can describe exponential growth and decay, the population growth of species or the change in investment return over time. A differential equation is one which is written in the form dy/dx = ………. Some of these can be solved (to get y = …..) simply by integrating, others require much more complex mathematics.

**Population Models
**

One of the most basic examples of differential equations is the Malthusian Law of population growth dp/dt = rp shows how the population (p) changes with respect to time. The constant r will change depending on the species. Malthus used this law to predict how a species would grow over time.

More complicated differential equations can be used to model the relationship between predators and prey. For example, as predators increase then prey decrease as more get eaten. But then the predators will have less to eat and start to die out, which allows more prey to survive. The interactions between the two populations are connected by differential equations.

The picture above is taken from an online predator-prey simulator . This allows you to change the parameters (such as predator birth rate, predator aggression and predator dependance on its prey). You can then model what happens to the 2 species over time. The graph above shows the predator population in blue and the prey population in red – and is generated when the predator is both very aggressive (it will attack the prey very often) and also is very dependent on the prey (it can’t get food from other sources). As you can see this particular relationship generates a population boom and crash – the predator rapidly eats the prey population, growing rapidly – before it runs out of prey to eat and then it has no other food, thus dying off again.

This graph above shows what happens when you reach an equilibrium point – in this simulation the predators are much less aggressive and it leads to both populations have stable populations.

There are also more complex predator-prey models – like the one shown above for the interaction between moose and wolves. This has more parameters to control. The above graph shows almost-periodic behaviour in the moose population with a largely stable wolf population.

Some other uses of differential equations include:

1) In medicine for modelling cancer growth or the spread of disease

2) In engineering for describing the movement of electricity

3) In chemistry for modelling chemical reactions

4) In economics to find optimum investment strategies

5) In physics to describe the motion of waves, pendulums or chaotic systems.

With such ability to describe the real world, being able to solve differential equations is an important skill for mathematicians. If you want to learn more, you can read about how to solve them here.

If you enjoyed this post, you might also like:

Langton’s Ant – Order out of Chaos How computer simulations can be used to model life.

Does it Pay to be Nice? Game Theory and Evolution. How understanding mathematics helps us understand human behaviour

**IB Revision**

If you’re already thinking about your coursework then it’s probably also time to start planning some revision, either for the end of Year 12 school exams or Year 13 final exams. There’s a really great website that I would strongly recommend students use – you choose your subject (HL/SL/Studies if your exam is in 2020 or Applications/Analysis if your exam is in 2021), and then have the following resources:

The Questionbank takes you to a breakdown of each main subject area (e.g. Algebra, Calculus etc) and each area then has a number of graded questions. What I like about this is that you are given a difficulty rating, as well as a mark scheme and also a worked video tutorial. Really useful!

The Practice Exams section takes you to ready made exams on each topic – again with worked solutions. This also has some harder exams for those students aiming for 6s and 7s and the Past IB Exams section takes you to full video worked solutions to every question on every past paper – and you can also get a prediction exam for the upcoming year.

I would really recommend everyone making use of this – there is a mixture of a lot of free content as well as premium content so have a look and see what you think.

## 11 comments

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August 30, 2014 at 2:34 am

LiamI’m interested in looking into and potentially writing about the modelling of cancer growth mentioned towards the end of the post, do you know of any good sources of information for this? is there anywhere that you would recommend me looking to find out more about it?

August 30, 2014 at 8:38 am

Ibmathsresources.comhiya

If you read the wiki page on Gompertz functions [http://en.wikipedia.org/wiki/Gompertz_function] this might be a good starting point. This function is a modified exponential model so that you have rapid initial growth (as in a normal exponential function), but then a growth slowdown with time. You could use this equation to model various initial conditions.

December 2, 2014 at 11:16 pm

NamiI was thinking of using related rates as my ia topic but Im not sure how to apply related rates into physics or medicine.

December 27, 2014 at 4:09 pm

sehrI was thinking of modelling traffic flow using differential equations, are there anything specific resources that you would recommend to help me understand this better?

August 11, 2016 at 12:54 am

Ekene ChidozieNice article

November 1, 2016 at 1:04 am

DHANANJAY KUMARVERY USEFUL

December 8, 2016 at 1:44 pm

LoganI have a paper due over this, thanks for the ideas!

January 12, 2018 at 9:46 pm

sanjay tiwariverry usefull

March 21, 2018 at 1:48 pm

archanavery nice article, people really require this kind of stuff to understand things better

March 9, 2019 at 3:44 pm

Kaleem Ur rehmanHow plz explain following?????

Application of differential equations?

) In medicine for modelling cancer growth or the spread of disease

2) In engineering for describing the movement of electricity

3) In chemistry for modelling chemical reactions

4) In economics to find optimum investment strategies

5) In physics to describe the motion of waves, pendulums or chaotic systems

July 22, 2020 at 2:51 pm

Dr Kay Khaing WinFirstly, l say that I would like to thank you. Actually, l would like to try to collect some facts to write a term paper for URJ . So l would like to study simple real problems solved by ODEs. Thank you.