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Module 1: Dynamics

Differential equations are the foundation of modern science. In this class, we'll approach the math gently, but we won't shy away from it either. In this section, we provide additional materials to help you familiarize yourself with or learn more about differential equations. Note that this section is intended to motivate you to learn about the topic. While these videos can help you gain some intuition, mathematics is fundamentally a know-how. To truly get it, you need to get your hands dirty.

Motivational videos

Additional readings

Courses

Interactive notebooks

Consolidating the basics

This section is for students looking to refresh their mathematical skills. They provide additional references to mathematical concepts that are so fundamental, it's important to have a good grasp of them.

Solving differential equations

Work in progress, there might be mistakes.

Even though we won't solve that many differential equations by hand, you will encounter the idea over and over again. Khan, Brillant, and Paul's online notes offer many exercices if you want to practices solving differential equations.

Here is my best shot at explaining the meaning of solving differential equations from scratch. Recall what solving mean in elementary school (yes, that far). When you solve for x, as in , you find a number. Here 7. At the risk of saying the obvious, you find the following

Later on, even if you don't want to remember it, you learn about derivative. For instance, the derivative is defined as a limit, or . Plugging in our function, we get . Later in the same course, you learn a shortcut; (the sum rule). Using the sum rule, and perhaps the Leibniz notation, you now think of derivatives as . You think to yourself; yes, I remember that.

Now, people say that solving differential equations mean that you solve for functions. This is where most people get lost. What does that mean? How is this idea related to modeling? This is best understood with an example. Say that we have the following relationship:

where is often use to talk about the size of a population at time . Take a second to understand what this equation means. The left hand side is a derivative, that is, the rate of change of at time . But the right hand side is not a number, it is "some function of ", here .

In the context of population growth, we are saying that the change in the population size is proportional to the size of the population itself. The parameter is the growth rate, i.e., how many offspring an individual produces (on average) per unit of time (notice has dimension equal to the inverse of time, say 1/seconds or 1/days). Multiplying by the current number of individuals in the population, namely , we get the rate of change of the whole population. Therefore, the bigger the population, the bigger the change. This is a modeling choice! As you will see in this module, we will use graph diagrams to represent those choices, e.g.,

I think it is worth reiterating; from the example above, you should see that (1) we have an unknown function and that (2) the latter and its derivative are connected by a relation . In this particular case, the educator will tell you something like; "think, what function do you know that is equal to its derivative?" Surprise, this is the exponential function:

Congrats, you just solved the mystery! What? WHY? How are we supposed to know that. Where is the coming from and what is its meaning? First, if you take in the equation above, given that , you find . That is, is the initial size of the population (one usually parametrizes time so that is the instant at which the system comes to exist, but it is just a convention; nothing deep about it), and so we can rewrite the solution explicitly as . Second, they say, you should remember that the exponential function is a function for which the rate of change at any point () is equal to itself, that is, (more generally, using the chain rule for derivatives, ). But it does feel like cheating.

In a "Differential equations" class, you learn to solve differential equations using various strategies. You learn that the above is a separable (first-order) differential equation, which means that it can be cast in the form . Our equation is the most trivial of this kind, for we have and (i.e., no explicit dependence on the independent variable, ). We thus proceed as follows (let us omit the dependence on to ease the notation),

To get to the second line, we used the definition of differential () and substituted the expression for (that we read from the first line). We then divided both sides by to have one side in terms of only, and the other in terms of only. To get the solution, we finally integrate (or anti-derivate if you prefer) on both sides:

defined , which is the same as above (see this refresher if you want to keep going)! In Otto and Day (p.24), they explain how derivatives and differential equations are related, which I think is often confusing to students:

Do you have more examples? Click on 'view Source' in the top right corner of the page, and propose your changes on the Git repository by clicking the ✎ icon.