SOLVING 1ODES WITH ELEMENTARY FUNCTIONS L.G.S. Duartea, L.A.C.P. da Motaa,∗, A.B.M.M. Queiroza,b aUniversidade do Estado do Rio de Janeiro, Instituto de F´ısica, Depto. de F´ısica Te´orica, 20559-900 Rio de Janeiro – RJ, Brazil bInmetro - Instituto Nacional de Metrologia, Qualidade e Tecnologia. Abstract Here we present a new approach to deal with first order ordinary differential equations (1ODEs) presenting functions. This method is an alternative to the one we have presented in [1, 2]. In [3], we have establish the theoretical background to deal with rational second order ordinary differential equations (2ODEs) (the S-function method). In this present paper, we combine the technique used in [3] with an idea analogous to that used in [1, 2] in order to produce a method that is much more efficient in a great number of cases. Directly, the appoach we present here for solving 1ODEs is applicable to any problem presenting parameters to which the rate of change is related to the parameter itself and, in addition, the theoretical results are on a more solid mathematical basis than those presented in [1, 2]. The method is implemented in a Maple package LeapS1ode and it includes commands that allow obtaining all the intermediate steps of the process of solving the 1ODEs. Keywords: 1ODEs with elementary functions, Polynomial vector fields, Liouvillian functions, S-function method arXiv:1710.00674v2 [math.CA] 10 May 2021 ∗Corresponding author L.G.S. Duarte and L.A.C.P. da Mota wish to thank Funda¸c˜ao de Amparo `aPesquisa do Estado do Rio de Janeiro (FAPERJ) for a Research Grant. Email addresses: [email protected] (L.G.S. Duarte), [email protected] or [email protected] (L.A.C.P. da Mota), [email protected] (A.B.M.M. Queiroz) Preprint submitted to Computer Physics Communications 11 de maio de 2021 PROGRAM SUMMARY Title of the software package: LeapS1ode – Liouvillian, elementary, algebraic and polyno- mial Solutions of 1ODEs (first order ordinary differential equations). Catalogue number: (supplied by Elsevier) Software obtainable from: CPC Program Library, Queen’s University of Belfast, N. Ire- land. Licensing provisions: none Operating systems under which the program has been tested: Windows 10. Programming languages used: Maple 17. Memory required to execute with typical data: 200 Megabytes. No. of lines in distributed program, including On-Line Help, etc.: 1037 Keywords: 1ODEs with functions, Polynomial vector fields, Liouvillian functions, S- function method. Nature of mathematical problem Search for general solutions of first order ordinary differential equations (1ODEs) with (elementary) functions. Methods of solution The method of solution is based on an algorithm described in this paper. Restrictions concerning the complexity of the problem If the 1ODE that is being analysed presents a very high degree in (x,y,θ), then the method may not work well. Typical running time This depends strongly on the 1ODE that is being analysed. Unusual features of the program Our implementation can find solutions in many cases where the 1ODE under study can not be solved by other powerful solvers nor by other powerful methods. Besides that, the program can perform (in a very natural way) an analysis of the integrability region of the 1ODE’s parameters. Finally, the package presents some useful research commands. 2 LONG WRITE-UP Apart from their mathematical interest, dealing with first order differen- tial equation (1ODEs) has its more direct application. Any physical (or, for that matter, any scientific question) that can be formulate as a relationship between the rate of change of some quantity of interest and the quantity itself can be formulated as solving an 1ODE (at some stage). There are many well known phenomena which fall into this category: Concentration and dilution problems are quickly remembered examples. Population models are another such example. More specifically, of non-linear 1ODEs and, as a final example, Astrophysics [4], etc. There are many others and there can be many more lurking around the corner of the many scientific endeavours being pursuit. In this sense, any novel mathematical approach, technique to deal with such 1ODEs are welcome and could prove vital to solve a funda- mental question. So, due to this great importance (theoretical and practical) of solving 1ODEs, many methods have been improved in the last decades. Regarding the search for elementary and Liouvillian solutions1, the methods that use the Darboux-Prelle-Singer approach stand out. In this ‘direction’ we can highlight: [5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 20]. We have been studying the existence of (and methods to search for) elementary and Liouvillian first integrals of vector fields in R2 since 2002 [1, 2, 12, 35, 36, 37]. In particular, we tackled the problem of looking for general solutions for 1ODEs with functions in [1, 2]. However, the mathema- tical basis was not laid out in such a solid way and, in addition, the method did not deal well (it was computationally expensive) with the problem of determining Darboux polynomials in three or more variables. This time, we managed to combine the technique we used in [3]2 with a reasoning very close to the idea used in [1, 2]3. Our method basically starts with an assumption (the existence of a general solution of a specific type) to arrive at an alge- braic result: the existence (or not) of a Liouvillian first integral (of a certain type) for a polynomial vector field in three variables and, in the case of a positive answer, the determination of that first integral (and, consequently, 1The methods that use the Lie symmetry approach have also been greatly improved. See, for example [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34]. 2In that work we use the (so-called) S-function (introduced in [38] and used by us and other people [3, 39, 40, 37, 41, 42, 44, 45, 46, 47]) that improves greatly the efficiency on the searching for Liouvillian first integrals of 2ODEs. 3The central idea is to ‘transform’ elementary functions into rational functions. 3 of the 1ODE’s general solution). Briefly, we associate the 1ODE (presenting an elementary function) with a polynomial vector field in three variables and then analyze the (possible) existence of a Liouvillian first integral. If the first integral is found we only have to perform the inverse transformation to obtain the general solution of the 1ODE in implicit form. This paper is organized as follows: In the first section, we present some basic concepts involved in the Darboux- Prelle-Singer (DPS) approach and in the S-function method. In the second section, we show how we can associate a 1ODE with an ele- mentary function to a polynomial vector field in three variables and develop a technique similar to the one we built in [3] (to search for Liouvillian first integrals of rational 2ODEs using the S-function method) in order to search for a Liouvillian first integral of the vector field. We use these two steps to propose a procedure that can succeed even in cases where the integrating fac- tors have Darboux polynomials of very high degree. In the end of this section (after a few examples in order to clarify the steps of the procedure), we pro- pose a semi algorithm to deal with solving 1ODEs presenting an elementary function and we describe it in a more formal way. In the third section, we present a computacional package (in Maple) to search for Liouvillian solutions of 1ODEs presenting an elementary function. We start by giving an overview of the package and presenting a summary of the commands. Then, we go deeper into the description of each command and, in the last subsection, we present an example of the usage of the commands to show how they work in practice. In the fourth section, we discuss the performance of the method. We begin by presenting a set of 1ODEs that the method solves without problems but, we believe, would not be easily solved by any other method. In the following, we present some special features of the package we use to solve / study more complicated cases. Finally, we show a case of a 2ODE (in principle, not treatable by the method) that the package can successfully handle. Finally, we present our conclusions. 1. Basic concepts and results In this section we will set the mathematical basis for presenting our method. In the first subsection we will present the Darboux-Prelle-Singer ap- 4 proach in a nutshell and, in the second, the basics of the S-function method. 1.1. The Darboux-Prelle-Singer approach When we talk succinctly about the Darboux-Prelle-Singer approach to se- eking Liouvillian first integrals of autonomous systems of polynomial 1ODEs in the plane, we can summarize the central idea in one paragraph: Find the Darboux polynomials (DPs) associated with the 1ODE system and use them to determine an integrating factor. Without exaggeration, the determination of the DPs is the most impor- tant and (unfortunately) the most complex part (by far) of the whole process. Thus, with a couple of definitions and results, we can describe (briefly) the DPS approach when applied to vector fields in the plane. To begin, let’s define more formally the concepts we referred to. Consider the following system of polynomial 1ODEs in the plane: x˙ = f(x, y) (1) ( y˙ = g(x, y), where f and g are coprime polynomials in (x, y) and the dot means derivative with respect to a parameter t (u ˙ ≡ du/dt). Definition 1.1. A function I(x, y) is called a first integral of the system (1) if I is constant over the solutions of (1).