Using Lie transformation groups to find closed form solutions to first order ordinary differential equations

摘要

Most work on computer programs to find closed form solutions to ordinary differential equations (o.d.e.s) has concentrated on implementing a catalog of those methods often cited in textbooks and reference works (see e.g. Kam61a, Inc44a]): algorithms of certain, easily recognized cases (e.g. separable, exact, homogeneous equations) and a useful guessing framework for changes of variable. This approach has been followed by Moses, Schmidt, and Lafferty, among others [Mos67a, Sch76a], [Sch79a, Laf80a].

We present here a different approach to cataloguing, using the relation between differential equations and Lie transformation groups. When presented with a given a first order o.d.e., we shall be concerned with finding continuous transformations (of the plane) which map the solution curves of the o.d.e. into each other. When a group of such transformations is found, it is possible to construct the solution to the o.d.e. via quadratures. We shall find that for many cases of interest, there are succinct algorithms for finding the transformations without knowing the solution curves beforehand. The guiding relationships for the catalogue search, and the justification for the quadrature formula used, has been known for about a century. Pioneering work was done by Sophus Lie and others in the 19th century (see e.g. [Lie75a]). Its emplacement within a symbolic/algebraic computational setting is, of course, modern-day.