Welcome to another academic year at Stevens! This will be my fourth year as a PhD student and eighth year in total at Stevens, and with it brings something new: This fall, I will be a teaching assistant for MA 221, a course on differential equations that many undergraduates take.
While the course material covers many techniques for solving differential equations and understanding their solutions, I thought I would write an article that gives a high level overview of the subject. I hope I can also convince you of the exciting nature of these equations, and their surprising ubiquity in real-world applications.
Let us start with a very basic question: What even is a differential equation? We have likely all seen mathematical equations at many points in our early education, but it’s equally likely that the solution to these equations was a number. For instance, the solution to the equation 3x + 1 = 7 would be x = 2.
However, with differential equations solutions are mathematical functions that are expressions that take different outputs based on given inputs. And the reason that differential equations are so interesting is because they describe a relationship between a function and that function’s derivatives.
Already, I have introduced a lot here. One can think of derivatives as an instantaneous rate of change of a function, but the term “instantaneous rate” is oxymoronic. Can we actually even trust a differential equation in telling us anything meaningful about a function? In fact we can, but to do so we must use the notion of limits to define derivatives. Limits themselves are hard to grasp intuitively. Think of Zeno’s paradox, where Achilles is racing a tortoise that starts a certain distance ahead. We expect Achilles to overtake the tortoise at some point, but Zeno describes this: Every time Achilles gets halfway closer to the tortoise, the tortoise itself has moved some distance away too! Describing this mathematically leads to the concept of limits and, with it, derivatives. But note that we are always on tricky footing if we really think about it!
Nonetheless, differential equations show up all around us. In their “simplest” form, the solutions of differential equations are functions of one variable, typically understood as time in modeling. With these “ordinary differential equations,” or ODEs, we can still model population growth of various species, the spread of communicable diseases, and the synchronization of flashes made by lightning bugs.
Even more exciting is when solutions are functions of multiple variables — we tend to call the corresponding equations “partial differential equations,” or PDEs. Here, we can model more complex processes, such as the propagation of electromagnetic waves, the conduction of heat through materials, and the dynamics of fluids, plasma, and other exotic phases of matter. These equations very quickly get very hard to solve!
Finally, one may argue that, while the above ODEs and PDEs may model various processes really well, there is always still some uncertainty in how we describe such complicated things. And indeed, one can introduce this uncertainty as specific types of randomness, which gives rise to “stochastic differential equations,” or SDEs. It would take at least another article to describe these faithfully, but suffice it to say these SDEs can be used to model financial markets, or in statistical mechanics, where one tries to make sense of the random motions of atoms and molecules.
To conclude, I hope this introduction to differential equations will also help quell some concerns that the content of MA 221 is useless or irrelevant. Even when the course material becomes extremely challenging, I find the myriad applications of such fundamental mathematical concepts to be equally motivating. And it may feel like we are chasing a tortoise just out of reach, but I have confidence in all of you to make significant academic strides this semester. I wish you a happy start to the Fall, and many successes in solving differential equations and related problems throughout!
