What are Differential Equations and how do they work?

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Today I want to talk about that piece of mathematics which describes, for all we currently know, everything: Differential Equations. Pandemic models? Differential equations. Expansion of the universe? Differential equations. Climate models? Differential equations. Financial markets? Differential equations. Quantum mechanics? Guess what, differential equations.


I find it hard to think of anything that’s more relevant for understanding how the world works than differential equations. Differential equations are the key to making predictions and to finding out what is predictable, from the motion of galaxies to the weather, to human behavior. In this video I will tell you what differential equations are and how they work, give you some simple examples, tell you where they are used in science today, and discuss what they mean for the question whether our future is determined already.

To get an idea for how differential equations work, let us look at a simple example: The spread of a disease through the population. Suppose you have a number of people, let’s call it N, which are infected with a disease. You want to know how N will change in time, so N is a function of t, where t is time. Each of the N people has a certain probability to spread the disease to other people during some period of time. We will quantify this infectiousness by a constant, k. This means that the change in the number of people per time equals that constant k times the number of people who already are infected.

Now, the change of a function per time is the derivative of the function with respect to time. So, this gives you an equation which says that the derivative of the function is proportional to the function itself. And this is a differential equation. A differential equation is more generally an equation for an unknown function which contains derivatives of the function. So, a differential equation must be solved not for a parameter, say x, but for a whole function.

The solution to the differential equation for disease spread is an exponential function, where the probability of infecting someone appears in the exponent, and there is a free constant in front of the exponential, which I called N₀. This function will solve the equation for any value of this free constant. If you put in the time t equals zero, then you can see that this constant N₀ is simply the number of infected people at the initial time.

So, this is why infectious diseases begin by spreading exponentially, because the increase in the number of infected people is proportional to the number of people who are already infected. You are probably wondering now how these constants relate to the basic reproduction number of the disease, the R naught we have all become familiar with. When a disease begins to spread, this constant k in the exponent is (R₀-1)/ τ, where τ is the time an infected person remains infectious.

So, R naught can be interpreted as the average number of people someone infects. Of course in reality diseases do not continue spreading exponentially, because eventually everyone is either immune or dead and there’s no one left to infect. To get a more realistic model for disease spread, one would have to take into account that the number of susceptible people begins to decrease as the infection spreads. But this is not a video about pandemic models, so let us instead get back to differential equations. Another simple example for a differential equation is one you almost certainly know, Newton’s second law, F equals m times a. Let us just take the case where the force is a constant. This could describe, for example, the gravitational force near the surface of the earth, in a range so small you can neglect that the force is actually a function of the distance from the center of Earth. The equation is then just a equals F over m, which I will rename to small g, and this is a constant. a is the acceleration, so the second time-derivative of position. Physicists typically denote the position with x, and a derivative with respect to time with a dot, so that is double-dot x equals g. And that’s a differential equation for the function x of t.

For simplicity, let us take x to be just the vertical direction. The solution to this equation is then x(t)= gt²/2 + vt +x₀, where v and x₀ are constants. If you take the first derivative of this function, you get g times t plus v, and another derivative gives just g. And that’s regardless of what the two constants were.

These two new constants in this solution, v and x₀, can easily be interpreted, by looking at the time t=0. x₀ is the position of the particle at time t = 0, and, if we look at the derivative of the function, we see that v is the velocity of the particle at t=0. If you take an initial velocity that’s pointed up, the curve for the position as a function of time is a parabola, telling you the particle goes up and comes back down. You already knew that, of course. The relevant point for our purposes is that, again, you do not get one function as a solution to the equation, but a whole family of functions, one for each possible choice of the constants.

Physicists call these free constants which appear in the possible solutions to a differential equation “initial values”. You need such initial values to pick the solution of the differential equation which fits to the system you want to describe. The reason we have two initial values for Newton’s law is that the highest order of derivative in the differential equation is two. Roughly speaking, you need one initial value per order of derivative. In the first example of disease growth, if you remember, we had one derivative and correspondingly only one initial value.

Now, Newton’s second law is not exactly frontier research, but the thing is that all theories we use in the foundations of physics today are of this type. They are given by differential equations, which have a large number of possible solutions. Then we insert initial values to identify the solution that actually describes what we observe.

Physicists use differential equations for everything, for stars, for atoms, for gases and fluids, for electromagnetic radiation, for the size of the universe, and so on. And these differential equations always work the same. You solve the equation, insert your initial values, and then you know what happens at any other moment in time.

I should add here that the “initial values” do not necessarily have to be at an initial time from which you make predictions for later times. The terminology is somewhat confusing, but you can also choose initial values at a final time and make predictions for times before that. This is for example what we do in cosmology. We know how the universe looks today, that are our “initial” values, and then we run the equations backwards in time to find out what the universe must have looked like earlier.

These differential equations are what we call “deterministic”. If I tell you how many people are ill today, you can calculate how many will be ill next week. If I tell you where I throw a particle with what initial velocity, you can tell me where it comes down. If I tell you what the universe looks like today, and you have the right differential equation, you can calculate what happens at every other moment of time. This consequence is that, according to the natural laws that physicists have found so far, the future is entirely fixed already; indeed, it was fixed already when the universe began.

This was pointed out first by Pierre Simon Laplace in 1814 who wrote:

“We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed, if this intellect were also vast enough to submit these data to analysis, it would embrace in a single formula the movements of the greatest bodies of the universe and those of the tiniest atom; for such an intellect nothing would be uncertain and the future just like the past would be present before its eyes.”

This “intellect” Laplace is referring to is now sometimes called “Laplace’s demon”. But physics didn’t end with Laplace. After Laplace wrote those words, Poincare realized that even deterministic systems can become unpredictable for all practical purposes because they are “chaotic”. I talked about this in my earlier video about the Butterfly effect. And then, in the 20th century, along came quantum mechanics. Quantum mechanics is a peculiar theory because it does not only use an differential equations. Quantum mechanics uses another equation in addition to the differential equation. The additional equation describes what happens in a measurement. This is the so-called measurement update and it is not deterministic.

What does this mean for the question whether we have free will? That’s what we will talk about next week, so stay tuned.