I have a love-hate relationship with calculus because I never really learned it properly, but I cannot live without it. I never quite liked the traditional approach to differential equations in most college courses since it involves a lot of technicalities, so I want to show you how I would have loved to study differential equations: not with the main purpose of becoming good at solving differential equations, but with the goal of describing the world around us using differential equations as our language. In the next three or four entries, I will tell you the story of a very curious girl called Relena (pronounced /rɛˈlənə/) as she discovers a new way to understand what happens around her.
But, first, let me explain why I say that I never learned all this stuff properly. The next four paragraphs are just some background about myself and you can safely skip them if you want to get right to Relena’s story.
The Prodigal Son
When I was a kid, people used to tell me that I was good at Mathematics (and I was, although I’m not sure if I can say that anymore). I believe that I wasn’t particularly talented at it but I just liked it. One of the most precious childhood memories I have with my dad is when he taught me how to convert numbers from decimal to binary (I know; super nerd alert). I always thought of maths as a game and I could spend the whole afternoon writing huge random numbers and converting them to binary, ternary and quaternary many years before I studied the concept of numeral systems at school. I kept learning many mathematical tools that I treasured in my brain like a kid treasures toys in a box. The result was that, all the way until high school, all the maths I learned at school seemed to be only a boring formalization of things I already knew.
Later on, my mom came across people who would encourage me to study Mathematics beyond what I could learn at school. This slowly drove me into participating in the National Mathematical Olympiad when I was 12, which, ironically, drove me into studying Chemistry later, but that’s another story. I kept participating in both Mathematics and Chemistry Olympiads until I managed to represent my country at the International Mathematical Olympiad and win a gold medal at the Ibero-American Chemistry Olympiad in the same year. For five years I focused on studying “Olympiad Mathematics” and Undergrad Chemistry simultaneously.
The odd thing is that, since I was always interested in simple and fun maths, and not so much in applied stuff, and we don’t study even the concept of limits in high school in my country, I never happened to study calculus. What you study for Math Olympiads doesn’t include calculus, either, since the focus is on creativity and not so much on functionality. To contribute to this recipe for disaster, the only place where I could get close to calculus was in Physical Chemistry, but, since not everyone had a strong mathematical background, we usually just memorized a bunch of formulae or explained things qualitatively, assuming that no one knew how to use integrals. The result: until I started college, although I had already learned how to derive and integrate by myself in order to study abroad, I always felt that using calculus in Physics and Chemistry was something very hard that I would learn some day in the distant future.
I can’t really explain the awkwardness I felt when I realized that I would study Thermodynamics from my second year at college only after having studied calculus and learning about differential equations. I went through several courses taught by applied mathematicians which only managed to kill my interest in mathematics with their formalisms and it was hard for me to see how what I learned was either fun or useful to a chemist, which were, at the time, the two criteria I used to decide whether something was worth studying. Luckily, I took a course on differential equations focused on Physical Chemistry and another online course by Boston University on edX which helped me realize the beauty and the importance of learning differential equations. I believe my situation was similar to that of many people who struggle to learn English because they need to pass a subject or get a good TOEFL score, whereas some people easily learn English just because they want to communicate with other English speakers and then English becomes not the goal, but the means. Similarly, I was learning differential equations in order to solve problems in an exam, when I could have noticed that it was a way to understand varied phenomena, from mechanics to population dynamics, chemical reactions to neural connections, and bridge to aircraft design.
Relena goes to the Aquarium
This time, I want to try an approach I really like. Assuming that we know how to derive and integrate, let’s see how we can slowly derive (no pun intended) several concepts regarding differential equations by first looking at simple phenomena that happen around us instead of going the other way around. In order to do so, let’s go on a field trip. We’re going to join our friend Relena on her visit to the city aquarium. This is no ordinary aquarium. Let’s just say it’s a bit… old.
Relena is a young girl who loves observing things moving or changing. She recently learned calculus and she realized that there is a mathematical way to express change: derivatives. She noticed that many problems about derivatives and integrals in her class were related to motion. For example, the derivative of the position of a certain object with respect to time, , tells us the rate at which that object moves. If that rate has a numerical value of 2, during the infinitesimally small period of time represented by , the object would move a distance twice as long as that of an object for which that derivative has a value of 1. Relena suddenly realizes that this rate of change in the position of an object is nothing but its velocity (this is, in fact, a more formal definition of velocity). Relena wonders if knowing that velocity is a derivative can help her understand different kinds of motion. Anyways, it’s too late to think about it since she has to sleep early to wake up on time to go to the aquarium the next day, so she decides to go to bed.
The next day, Relena wakes up and takes a bus to the aquarium. There is no much traffic on the road so the bus is going at an almost constant speed (60 Km/h). Although the equations that rule uniform linear motion are trivial, Relena wonders how we could deal with it just knowing that there are no forces acting on the bus (first law of Newton) and thinking about derivatives. That is:
So, what is acceleration? Relena knows that it’s the rate of change of velocity, so she writes:
This is Relena’s first differential equation. Relena notices that she has a derivative on one side and an expression that can be easily integrated on the other side, so she integrates:
where A is the constant of integration. Relena wonders why she gets only a constant which could take any value. She realizes that the first equation () holds true for any bus moving at any constant speed, so it makes sense that we don’t get a specific answer but a constant whose value will be decided by the specific situation we are in. Relena then simply uses the fact that, at an arbitrary time, , therefore, at any time.
Relena wants to go further because she knows that velocity is the rate of change of position. Since the bus has been moving on the same street in the same direction, we can focus only on one coordinate, namely x, and repeat the same process:
where B is the constant of integration. Relena wonders now what this B term means. This time, however, we have a term that depends on time in our equation. Relena tries evaluating x for t=0:
That’s it! B is the position of the bus at our chosen initial moment! The bus is now about 2 Km away from Relena’s house. If we chose t=0 to be this exact moment and x to be the distance from Relena’s house, we can describe the motion of the bus with the equation:
“Well,” Relena thinks, “I could have easily derived this without any derivatives.” She is right, but she has actually just opened the door to a different way of approaching physical phenomena. Let’s analyze what Relena did. She solved first a differential equation for velocity and then for position. But she could’ve done it all in one step if she had substituted velocity for the derivative of position with respect to time in the first place:
So far we had encountered only differential equations with a first derivative, but this one has a second derivative. Such equations are called first-order and second-order differential equations, respectively. In this case, it is not such a big deal since we can still simply integrate both sides twice with respect to t and we would get:
This is exactly the kind of equation that Relena obtained. We have two constants here, so we need to know two pairs of values for in order to get a system of two equations that will allow us to find them. We can see that when (that would be the first pair, ), and once we know B, if we are given another pair , we can calculate and obtain Relena’s solution. We can conclude from this that every time we solve a differential equation, we will be left with constants or parameters that we can determine only if we know specific solutions of the equation. Regarding these solutions as “initial conditions”, we call problems of this kind initial value problems.
Relena finally arrives at the aquarium, which looks a little bit old. As she enters, a drop of water falls on her shoulder. She looks up and sees a leaking pipe. Before realizing that a water leakage in an aquarium can’t be a good thing, she starts thinking about the falling motion of the drop. It should be similar to the bus problem, but this time the acceleration is not zero but g. She is about to write down another differential equation on a napkin she found in her pocket, but she drops it. She observes the napkin slowly falling as the wind underneath offers its resistance, and she realizes that this is a more interesting problem.
Will Relena be able to solve this new problem? You can subscribe to this blog by clicking the “follow” button on the top right corner (or below if you’re using a smartphone), or you can also follow The Relearner on Facebook, Twitter, and Instagram to see our latest posts right on your news feed. The Relearner works for you and also thanks to you, so please check the Support Us section of our blog. Share this entry on Facebook, Twitter or Instagram with the hashtag #TheRelearner and tell us any situation in your everyday life from which you can obtain a differential equation. You’ll have the chance to get a souvenir from Nagoya University. A winner will be chosen every month. Thank you and see you next time!