Lie symmetry method

Summary

The Lie symmetry method solves differential equations by using their continuous symmetries. A Lie point symmetry is a one-parameter group of changes of variables that maps solutions to solutions. In adapted coordinates, called canonical coordinates, the symmetry becomes a translation and the equation can be integrated by quadrature. The method, due to Sophus Lie, explains most of the classical tricks for first-order equations, such as the substitution $y = x v$ for a homogeneous first-order equation. See the definition.