Solving second-order differential equations with Lie symmetries

Lie's theory for solving second-order quasilinear differential equations ba sed on its symmetries is discussed in detail. Great importance is attached to constructive procedures that may be applied for designing solution algor ithms. To this end Lie's original theory is supplemented by various results that have been obtained after his death one hundred years ago. This is tru e above all of Janet's theory for systems of linear partial differential eq uations and of Loewy's theory for decomposing linear differential equations into components of lowest order. These results allow it to formulate the e quivalence problems connected with Lie symmetries more precisely. In partic ular, to determine the function field in which the transformation functions act is considered as part of the problem. The equation that originally has to be solved determines the base field, i.e. the smallest field containing its coefficients. Any other field occurring later on in the solution proce dure is an extension of the base field and is determined explicitly. An equ ation with symmetries may be solved in closed form algorithmically if it ma y be transformed into a canonical form corresponding to its symmetry type b y a transformation that is Liouvillian over the base field. For each symmet ry type a solution algorithm is described, it is illustrated by several exa mples. Computer algebra software on top of the type system ALLTYPES has bee n made available in order to make it easier to apply these algorithms to co ncrete problems.