# Real world applications of differential equations

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• greg_rack
In summary: Hi guys, In summary, my maths teacher asked me to work on and deliver an engaging insight-oriented "lesson" to my class, about physical/engineering and real-world applications of differential equations, in order to better get the meaning of operating with such mathematical objects. Of course, growth and decay phenomena, along with motion equation(2nd order diff. eq.) came to my mind... but I was wondering if I might have been able to delve a bit deeper into the subject by delivering more "advanced" and interesting examples of such applications(even if still doable with high-school calculus knowledge). Online, I only found the brachistochrone problem which particularly caught my interest
greg_rack
Gold Member
Hi guys,
how are you doing?

My maths teacher asked me to work on and deliver an engaging insight-oriented "lesson" to my class, about physical/engineering and real-world applications of differential equations, in order to better get the meaning of operating with such mathematical objects.

Of course, growth and decay phenomena, along with motion equation(2nd order diff. eq.) came to my mind... but I was wondering if I might have been able to delve a bit deeper into the subject by delivering more "advanced" and interesting examples of such applications(even if still doable with high-school calculus knowledge).
Online, I only found the brachistochrone problem which particularly caught my interest, and which seemed not too advanced at a glance... but then I found out it involved concepts such as the Euler-Lagrange equation, which is definitely too beyond my class' level.

Have you got other ideas? Or should I stick just to decay and motion?

It's hard to find a field in science or engineering that doesn't use differential equations. I'll just name a few: (1) design of electronic circuits like in your cell phone and computer, (2) design of virtually any building or mechanical structure, (3) modeling and control of the electrical grid, (4) modeling of stars and galaxies, (5) modeling of cosmology, (6) modeling of population dynamics, (7) modeling of pandemics (like COVID-19), (8) predicting the weather, (9) modeling climate change ... I could go on, but maybe that's enough

greg_rack, russ_watters and Delta2
Lotka-Volterra should be doable. Here is an example:

And here is Kepler's law, albeit a bit artificial:

I have another example with Noether's theorem, but this might be too advanced.

greg_rack
greg_rack
The most interesting phenomenon of differential equations to me is how they model abrupt changes which are not uncommon in Nature, like the straw which breaks the camel's back: You would think just fishin' a bit wouldn't hurt too much; they'll quickly reproduce. But it depends where the population is. Imagine just removing several and the entire population collapses, basically going extinct. Take for example the DE which models harvesting of natural resources:
$$\frac{dP}{dt}=kP\left(1-\frac{P}{N}\right)-C$$
This DE is a function of the parameter ##C##, the "catch"of fish per season. As the catch increases, the population decreases a bit. But at some point, a bifurcation point, if the catch increases just a tiny bit more, the population collapses.

If interested, try and get a copy of "Differential Equations" by Blanchard, Devaney and Hall. Nice write-up about this. Try and use Mathematica or another to make some plots showing this phenomenon graphically. Very easy to solve the equation above numerically in Mathematica using NDSolve and plotting the results without having to know all the details of DE theory. Then maybe research other phenomena which exhibit phase-transitions, critical-points, bifurcations, and catastrophe points.

greg_rack
Hitting a moving target with whatever you're throwing at it.

greg_rack

## 1. What are some examples of real world applications of differential equations?

There are many examples of real world applications of differential equations, including modeling population growth, predicting weather patterns, designing electrical circuits, and analyzing the spread of diseases.

## 2. Why are differential equations important in science and engineering?

Differential equations are important in science and engineering because they provide a mathematical framework for describing and predicting the behavior of complex systems. They allow us to model and understand phenomena that cannot be easily observed or measured.

## 3. How are differential equations used in physics?

Differential equations are used extensively in physics to describe the motion of objects, the behavior of fluids, and the interactions between particles. They are also used to model and understand phenomena such as heat transfer, quantum mechanics, and electromagnetism.

## 4. Can differential equations be solved analytically or only numerically?

Some differential equations can be solved analytically, meaning that an exact solution can be found using mathematical methods. However, many differential equations do not have analytic solutions and must be solved numerically using computational methods.

## 5. How do differential equations contribute to the field of machine learning?

Differential equations are used in machine learning to model and predict complex systems, such as stock market fluctuations or human behavior. They are also used in the development of algorithms for pattern recognition and data analysis.

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