Mathematics Seminar - Ivanova Kseniya, Alpha Romeo F1-team, Hinwil, Switzerland

Title: Linear Inhomogeneous Differential Equations with Constant Coefficients and its applications to Mechanics

Abstract:

In this lecture, we investigate the solution techniques for the inhomogeneous linear differential equations (DEs) with constant coefficients. We start by swiftly revisiting how to find the general

solution of homogeneous DEs . Then, we consider inhomogeneous case and how the particular solutions are selected in the special cases that come up most often in physics . Our intellectual journey culminates with the practical application of our newfound knowledge to the realm of driven harmonic oscillators – systems influenced by external forces. As an example, we will solve the following second-order differential equation:

mx ̈(t) + cx ̇(t) + kx(t) = F cos(ωt) ,

where m, c and k stand for the mass, damping coefficient and stiffness of the linear system and F and ω represent the amplitude and the angular frequency of the exciting force. In the end, we will analyze the collapse of Tacoma Narrows Bridge. The main bridge span falling into the strait on November 7, 1940.

Join us as we navigate this captivating mathematical landscape, unveiling the harmony between theory and real-world physics.

Keywords— inhomogeneous linear differential equations, method of undetermined coefficients, driven harmonic oscillators, the collapse of Tacoma Narrows Bridge References

[1] Boyce W.E., DiPrima R.C., Meade D.B. Elementary Differential Equations and Boundary Value Problems

[2] S. K. Godunov, Ordinary Differential Equations with Constant Coefficient

[3] https://en.wikipedia.org/wiki/Tacoma_Narrows_Bridge_(1940)#Collapse