I gave the talk "Using symmetry to solve differential equations" at Lewis and Clark College on March 6, 2012. The talk was aimed at undergraduate students. My main goal was to give an overview of the main ideas in using Lie symmetries to solve differential equations in order to provide a jump start to someone wanting to study a book such as Hydon's Symmetry Methods for Differential Equations: A Beginner's Guide I also wanted to provide some dynamic visualizations that are hard to include in a printed text. Versions below use Wolfram's Computable Document Format. These visualizations require a browser plugin that comes with Wolfram's free CDF Player or Mathematica.
This visualization shows a one-parameter family of transformations (or "transformation flows") acting on the plane. The transformation flow is visualized by the effect on a grid of points. The parameter can be adjusted with the slider. The drop-down menu gives a variety of transformations.
This visualization shows a one-parameter family of transformations (or "transformation flows") acting on a circle in the plane. The parameter can be adjusted with the slider. The drop-down menu gives a variety of transformations.
This visualization shows how scaling in one direction affects slope. The scaling parameters, one for the horizontal direction and one for the vertical direction, can be adjusted with the sliders.
This visualization shows the effect of various transformation flows on the slope field for the differential equation dy/dx=y.
This visualization shows the effect of various transformation flows on the slope field for the differential equation dy/dx=y/x+y2/x3.