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Visualizing and Utilizing Symmetries of Differential Equations

Shirley Yap (California State University East Bay)

Abstract.  The symmetry method is a beautiful and powerful theory that sheds light on the myriad of seemingly unrelated techniques presented in elementary differential equations classes. The method, pioneered by Sophus Lie in the latter part of the nineteenth century, uses the invariance of the equation under certain transformations to create a coordinate system in which the equation greatly simplifies. For example, the technique transforms any first-order ordinary differential equation with a continuous family of symmetries into a separable equation. Most texts on symmetries of differential equations target a more mathematically advanced audience with knowledge of Lie groups and differential geometry. In this paper, we present the subject in an elementary and visual way, taking advantage of the online format to highlight the geometric nature of the subject with many animations and graphics.

Table of Contents

Shirley Yap (California State University East Bay), "Visualizing and Utilizing Symmetries of Differential Equations," Convergence (August 2016)