Department of Mathematics and Computer Science

California State University East Bay

Hayward, CA

shirley.yap@csueastbay.edu

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.

- Abstract
- 1. Introduction
- 2. Symmetries of Algebraic Equations
- 3. Symmetries of Differential Equations
- 4 The Geometry of Prolongations
- 5. Translational Symmetries and Orbits
- 5.1 Solving ODEs with Translational Symmetries
- 5.2 Orbits of Symmetries
- 5.3 Orbits of $\frac{dy}{dx}=xy^2 - \frac{2y}{x}-\frac{1}{x^3}$
- 5.4 Orbits of $\frac{dy}{dx}=\frac{2}{x^2}-y^2$
- 6. Linearizing the Symmetry Condition
- 7. Canonical Coordinates and Solving the ODE
- 8. Symmetries of Higher Order Differential Equations
- 9. Symmetry and Some Standard Techniques
- 10. Conclusion
- 11. References