Solving Differential Equations Using Symmetry Methods

In the late nineteenth century, Sophius Lie developed a technique to solve differential equations using symmetries of solutions. Briefly, a symmetry of a differential equation is a transformation that sends solutions to solutions. In this paper, we explore the connection between a first order differential equation and a corresponding one parameter (local) Lie group of symmetries. Symmetries of a differential equation are then used to find a change of variables (i.e., canonical coordinates) where solving becomes a standard integration problem. We also consider using symmetries to develop an integrating factor which makes our differential equation exact and thus easily solved. Ultimately we leverage symmetries to solve complicated first order differential equations that could not be solved otherwise.