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December 4, 2023

Nathan Pennington highlights the limitation of exact answers in Ex Numera lecture

By ZACHARY BAHAR | October 18, 2023



Nathan Pennington discussed the problems with the way differential equations are presented in courses.

Associate Professor in the Department of Mathematics at Creighton University Nathan Pennington delivered a guest lecture to Ex Numera, the undergraduate mathematics club, on Oct. 9. It was their third speaker of the semester. The talk, titled “Why You Should Take Differential Equations,” discussed the issues with typical first-semester differential equations courses in comparison to what the topic looks like.

Differential equations are equations involving a set of functions and their derivatives or rates of change. For example, a particularly famous differential equation is Newton’s Second Law of Motion, F=ma (force is equal to mass times acceleration), where acceleration can be expressed as the second derivative of position with respect to time. In practice, differential equations are often used to model the relationships in changing systems and appear throughout the sciences.

A first-semester differential equations class — including those offered at Hopkins — begins by building upon the foundations laid in Calculus II by focusing on simple differential equations and considering a variety of methods used to evaluate them, including separation of variables, integrating factors and exact equations before moving onto solving systems of equations. Each of these processes is formulaic, suggesting that differential equations as a whole are rote; according to Pennington, this is far from the case.

“A first-semester class focuses on solving differential equations using a formula box of some sort, it inevitably involves integration, you play a little game, do some terrible integrals and your homework problems almost invariably just involve e’s and sin’s so that repeated use of integration by parts gives an answer,” he said. “It's what we have to do, but it does tend to leave the mistaken impression that this is what doing differential equations is about. The intent of those examples is to point out that these techniques have very limited actual use because your chances of getting those integrals and getting actual analytic solutions are very low.”

As Pennington notes, most differential equations can’t be expressed analytically using standard functions often encountered in a calculus class. Even integrals of simple combinations of functions, like e^{x^3}sin(x), cannot typically be expressed in this way. Furthermore, the types of solutions offered by these methods often seem out of touch with the problems that produce them.

As an example, the population of bacteria on a petri can be modeled using the equation dy/dt=ky; the solution to this equation is an exponential function. This presents two challenges. First of all, the function does not return integer values and it grows towards infinity as time approaches infinity. The second of these problems can be resolved by adjusting our initial equation to account for environmental factors. 

However, according to Pennington, the first speaks to a deeper problem underlying the teaching of pure math versus teaching math in an applied setting: artificial precision.

“Part of the bridge from doing math in math classes to doing math outside of math classes is the notion of artificial precision. Do you think anybody really cares that the population you're predicting is 1672.5 instead of 1600? Do you think they can count the population with that level of accuracy? Of course not,” Pennington said. “Nothing, when you start trading with real life, has accuracy the way mathematics does. You wouldn't care that it's not giving you an integer — that was never an option in the first place — you're just trying to get close.”

This observation led Pennington to assert that the essence of differential equations, an aspect that needs to be prioritized in introductory courses, is qualitative. In the real world, understanding the basic structure of a system is more useful than attempting to find a (potentially non-existent) exact solution.

One means of gaining this understanding is by using the existence and uniqueness theorems for solutions to differential equations to draw diagrams that depict families of solutions based on the equilibrium solutions — constant solutions — of the system. This gives general descriptions rather than exact expressions, which may seem troubling to many who expect such solutions.

“That is not the way things are typically performed in a math context. Does it matter? Matter is a tricky word. It matters if you want all your derivations to be exactly correct, but when you're doing these more qualitative approaches, what you want to know is if your system reflects what should happen and if you get close enough to that to make predictions that are testable,” he said.

This qualitative nature enables differential equations to be applied to fields ranging from biology to physics and from economics to chemistry. They enable the study of the general behavior of a system over time with proven results, even and especially if there is no precise solution.

Further, studies of differential equations apply to a wide variety of mathematical concepts beyond the realm of simple functions, offering a way of uncovering patterns underlying seemingly disconnected topics.

“Differential equations are not about integrating things, it’s about structure — it’s about analyzing the behavior of stuff, even if you don't know exactly what it is. You notice there's a lot of linear algebra, you could also do this with abstract algebra: Do you want to deal with functions on groups, rings or fields? You can do differential equations on those. Do you want to integrate a group? That exists,” Pennington said. “If you try hard, everything is a [differential equations] problem.”

President of Ex Numera Yoyo Jiang shared her thoughts on the event in an interview with The News-Letter.

“It was great because we had our first analysis speaker of the semester. I hope people take away that differential equations aren’t boring and that there are useful aspects of them,” she said. “It’s good when we're touching on a bunch of different things that people actually do when using differential equations and not just looking at integration by parts over and over again.”

Assistant Dean for Undergraduate Academic Affairs and Associate Teaching Professor Emily Braley expressed her excitement for the event, stressing the importance of having undergraduate events in-person again after a hiatus due to COVID-19.

“The department is really glad to have the math club back to in-person meetings, bringing in speakers and bringing so much energy,” she said in an interview with The News-Letter. “It’s one of the highlights of being here, being in this department.”

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