Using Invariants to Solve the Equivalence Problem for Ordinary Differential Equations

Abstract

Two given ordinary differential equations (ODEs) are called equivalent if one can be transformed into the other by a change of variables. The equivalence problem consists of two parts: deciding equivalence and determining a transformation that connects the ODEs. Our motivation for considering this problem is to translate a known solution of an ODE to solutions of ODEs which are equivalent to it, thus allowing a systematic use of collections of solved ODEs. In general, the equivalence problem is considered to be solved when a complete set of invariants has been found. In practice, using invariants to solve the equivalence problem for a given class of ODEs may require substantial computational effort. Using Tresse's invariants for second order ODEs as a starting point, we present an algorithmic method to solve the equivalence problem for the case of no or one symmetry. The method may be generalized in principle to a wide range of ODEs for which a complete set of invariants is known. Considering Emden-Fowler Equations as an example, we derive algorithmically equivalence criteria as well as special invariants yielding equivalence transformations.