Simplifying Mathematical Processes in Differential Equations

Ordinary differential equations (ODEs) is a topic that many university students who have taken a mathematics course will be familiar with. For some, it will be a beloved memory, for others, it may have been a struggle. A University of Guelph professor has introduced a new simpler method to solving resonantly forced linear ODEs. This new method can be taught to undergraduate students to improve student experience and learning. 

ODEs In Real World Application 

ODEs also play a role in various other scientific disciplines. They have a role in computer science for algorithm design and proofs, and in disease and growth modeling for epidemiology. From a physics perspective, these equations are used to calculate the movement and flow of electricity, in thermodynamics concepts, and to calculate movements of objects such as a pendulum.  

The idea of resonance is typically a physics-based concept, while ODEs is a term that mathematicians are familiar with. Resonance is a concept that occurs in the “natural world” and is defined as a situation where a periodic force is applied to an object at its natural frequency causing it to vibrate at a higher amplitude. Resonance was the phenomenon behind the collapse of the Tacoma Narrows Bridge in 1940. Due to the wind blowing, the bridge oscillated (swung back and forth) at a frequency that ultimately caused the collapse. Understanding concepts such as resonance and ODEs is imperative  for both scientists in various industries as well as mathematicians. 

Simplifying the Math 

A recent article in the education section of the SIAM review, written by Gouveia and Stone, proposed a new method for solving resonantly forced linear ODEs. This approach was applied to seven examples to prove the method was correct. University of Guelph (U of G) Department of Mathematics and Statistics professor Dr. Allan Willms took it one step further by simplifying the method proposed. This was done by using the mathematical justification of the first article (by Gouveia and Stone) directly, which allows the mathematician to avoid unnecessary and tedious mathematical steps. Dr. Willms used his new method to solve the original seven examples and received the same answer with less steps. 

“It could be introduced into an undergraduate math course here at Guelph, and then whoever reads the article might want to introduce it at their university as well. It’s a nice method. It’s pretty straight forward and its main advantage is that it’s a lot faster,” said Willms. 

This story was written by Izabela Savić as part of the Science Communicators: Research @ CEPS initiative. Izabela is a MSc candidate in the School of Computer Science under Drs. Xiaodong Lin. Her research focus is on network security and computer forensics to better equip law enforcement and user security. 

Reference: A. Willms, “Resonantly Forced ODEs and Repeated Roots,” SIAM Review, vol 66, pp. 149-160, 2024.