Solution of Ordinary Differential Equations with MathLie

This article discusses Lie’s method of canonical variables to solve ordinary differential equations. The method of canonical variables is based on point symmetries and allows to construct transformations which simplify the equation prior to its solution. The method of canonical variables is closely related to the methods of first integrals and the method of first order partial differential equations. We discuss the necessary tools, the skeleton and the class of solution, providing the solution in connection with computer algebra. The procedure of canonical variables is algorithmic and implemented in MathLie. We demonstrate the application of the method on first- and second-order ordinary differential equations.