Elementary first integrals of differential equations

Abstract

Abstract. We show that if a system of differential equations has an elementary first integral (i.e. a first integral expressible in terms of exponentials, logarithms and algebraic functions) then it must have a first integral of a very simple form. This unifies and extends results of Mordukhai-Boltovski, Ritt and others and leads to a partial algorithm for finding such integrals.

Darboux integrability and invariant algebraic curves for planar polynomial systems

“…In [4,9,10,20], the authors developed the Darboux theory of integrability essentially in R 2 or C 2 considering not only the invariant algebraic curves but also the exponential factors, the independent singular points and the multiplicity of the invariant algebraic curves. Prelle and Singer [25], using methods of differential algebra, showed that if a polynomial vector field has an elementary first integral, then it can be computed using the Darboux theory of integrability. Singer [26] proved that if a polynomial vector field has Liouvillian first integrals, then it has integrating factors given by Darbouxian functions.…”

“…Substituting these last two equations into (31), we obtain that system (1) has the form (25) with the l + 1 invariant algebraic curves C 1 , . .…”

Solutions of some second order ODEs by the extended Prelle-Singer method and symmetries

“…In 1983 Prelle and Singer [1,2] devised a procedure which could not only determine polynomial first integrals but more importantly could be applied to systems admitting rational first integrals.…”

“…On the other hand for first-order scalar ODEs, S. Lie devised a method for constructing an integrating factor from each admitted point symmetry. Then, after almost a century, a major breakthrough in the construction of an algorithm for solving first-order ordinary differential equations was put forward by Prelle and Singer [1] in 1983. The method is a semi algorithmic procedure for solving nonlinear first-order ordinary differential equations of the form dy dx = P (x, y)…”

Applications of MetiTarski in the Verification of Control and Hybrid Systems

“…3 We are planning to extend our case studies to cover nonlinear cases by finding methods of solving systems of polynomial nonlinear ordinary differential equations analytically in terms of elementary and special functions. An example of such method is the Prelle-Singer procedure [12], extensions of which are also implemented in computer algebra systems such as REDUCE (the PSODE package [13]) and Maple (the PSsolver package [8]). …”