Related papers: First order ODEs, Symmetries and Linear Transforma…
A set of Maple V R.3/4 computer algebra routines for the analytical solving of 1st. order ODEs, using Lie group symmetry methods, is presented. The set of commands includes a 1st. order ODE-solver and routines for, among other things: the…
This paper is centred on solving differential equations by symmetry groups for first order ODEs and is in response to Starrett (2007). It also explores the possibility of averting the assumptions by Olver (2000) that, in practice finding…
This study will explicitly demonstrate by example that an unrestricted infinite and forward recursive hierarchy of differential equations must be identified as an unclosed system of equations, despite the fact that to each unknown function…
An alternative proof of Lie's approach for linearization of scalar second order ODEs is derived using the relationship between $\lambda$-symmetries and first integrals. This relation further leads to a new $\lambda$-symmetry linearization…
We discuss the role and merits of symmetry methods for the analysis of biological systems. In particular, we consider systems of first order ordinary differential equations and provide a comprehensive review of the geometrical foundations…
While quantum computing provides an exponential advantage in solving linear differential equations, there are relatively few quantum algorithms for solving nonlinear differential equations. In our work, based on the homotopy perturbation…
A systematic algorithm for building integrating factors of the form mu(x,y), mu(x,y') or mu(y,y') for second order ODEs is presented. The algorithm can determine the existence and explicit form of the integrating factors themselves without…
An update of the ODEtools Maple package, for the analytical solving of 1st and 2nd order ODEs using Lie group symmetry methods, is presented. The set of routines includes an ODE-solver and user-level commands realizing most of the relevant…
A systematic algorithm for building integrating factors of the form mu(x,y') or mu(y,y') for non-linear second order ODEs is presented. When such an integrating factor exists, the algorithm determines it without solving any differential…
In this set of papers we formulate a stand alone method to derive maximal number of linearizing transformations for nonlinear ordinary differential equations (ODEs) of any order including coupled ones from a knowledge of fewer number of…
A characterization of the symmetry algebra of the $n$th order ordinary differential equations (ODEs) with maximal symmetry and all third order linearizable ODEs is given. This is used to show that such an algebra $\mathfrak{g}$ determines…
Transformations of differential equations to other equivalent equations play a central role in many routines for solving intricate equations. A class of differential equations that are particularly amenable to solution techniques based on…
We introduce basic aspects of new operator method, which is very suitable for practical solving differential equations of various types. The main advantage of the method is revealed in opportunity to find compact exact operator solutions of…
There exist sound literature and algorithms for computing Liouvillian solutions for the important problem of linear ODEs with rational coefficients. Taking as sample the 363 second order equations of that type found in Kamke's book, for…
We consider a class of linear ODEs of second order with variable coefficients and construct its Lie algebra of Lie group of equivalence transformations. Further we find invariants and differential invariants of this Lie algebra and by using…
Whereas Lie had linearized scalar second order ordinary differential equations (ODEs) by point transformations and later Chern had extended to the third order by using contact transformation, till recently no work had been done for higher…
In this paper, we give an algorithm for finding general rational solutions of a given first-order ODE with parametric coefficients that occur rationally. We present an analysis, complete modulo Hilbert's irreducibility problem, of the…
A straightforward algorithm for the symbolic computation of higher-order symmetries of nonlinear evolution equations and lattice equations is presented. The scaling properties of the evolution or lattice equations are used to determine the…
A new class of vector fields enabling the integration of first-order ordinary differential equations (ODEs) is introduced. These vector fields are not, in general, Lie point symmetries. The results are based on a relation between…
This article is a study about the existence and the uniqueness of solutions of a specific quadratic first-order ODE that frequently appears in multiple reconstruction problems. It is called the \emph{planar-perspective equation} due to the…