Related papers: First order algebraic differential equations of ge…
First order algebraic differential equations are considered. An necessary condition for a first order algebraic differential equation to have a rational general solution is given: the algebraic genus of the equation should be zero.…
In this paper, we mainly investigate on the finite order transcendental entire solutions of two Fermat types delay-differential and one Fermat type c-shift equations, as these types were not considered earlier. Our results improve those of…
Let $k$ be a differential field of characteristic zero with an algebraically closed field of constants. In this article, we provide a classification of first order differential equations over $k$ and study the algebraic dependence of…
For a second order linear differential equation $f''+A(z)f'+B(z)f=0$, with $ A(z)$ and $B(z)$ being transcendental entire functions under some restriction, we have established that all non-trivial solutions are of infinite order. In…
The problem of algebraic dependence of solutions to (non-linear) first order autonomous equations over an algebraically closed field of characteristic zero is given a `complete' answer, obtained independently of model theoretic results on…
In this paper, we study all possible orders which are less than 1 of transcendental entire solutions of linear difference equations \begin{equation} P_m(z)\Delta^mf(z)+\cdots+P_1(z)\Delta f(z)+P_0(z)f(z)=0,\tag{+} \end{equation} where…
We propose an algebraic geometric approach for studying rational solutions of first-order algebraic ordinary difference equations. For an autonomous first-order algebraic ordinary difference equations, we give an upper bound for the degrees…
In this paper, we study about existence and non-existence of finite order transcendental entire solutions of the certain non-linear differential-difference equations. We also study about conjectures posed by Rong et al. and Chen et al.
This manuscript presents a novel and reliable third-order iterative procedure for computing the zeros of solutions to second-order ordinary differential equations. By approximating the solution of the related Riccati differential equation…
Arithmetic differential equations are analogues of algebraic differential equations in which derivative operators acting on functions are replaced by Fermat quotient operators acting on numbers. Now, various remarkable transcendental…
In this paper, we study the growth of transcendental entire solutions of linear difference equations \begin{equation} P_m(z)\Delta^mf(z)+\cdots+P_1(z)\Delta f(z)+P_0(z)f(z)=0,\tag{+} \end{equation} where $P_j(z)$ are polynomials for…
A generalization of the already studied transformations of the linear differential equation into a system of the first order equations is given. The proposed transformation gives possibility to get new forms of the N-dimensional system of…
In this paper, the exact solutions of certain non-linear differential equations defined on a fractal subset of the real line are presented. Particular attention is paid to the Riccati-type fractal differential equation, for which a…
In this paper we study properties of regular solutions of quaternionic Riccati equations. The obtained results we use for study of the asymptotic behavior of solutions of two first-order linear quaternionic ordinary differential equations.
We show how to approximate a solution of the first order linear evolution equation, together with its possible analytic continuation, using a solution of the time-fractional equation of order $\delta >1$, where $\delta \to 1+0$.
We consider properties of binomial series $\sum_{n=0}^\infty a_n z^{\underline{n}}$, where $z^{\underline{n}}=z(z-1)\cdots(z-n+1)$ and the convergence of binomial series in the complex domain. The order of growth of entire and meromorphic…
In this paper we present a direct formula for the solution of the general second order linear ordinary differential equation as our main result such that the parameters required for the formula are determined using another differential…
The Riccati equation method is used for study the behavior of solutions of the systems of two linear first order ordinary differential equations. All types of oscillation and regularity of these system are revealed. A generalization of…
There is the paper by H. Tietze published in 1905 on differential transcendence of solutions of difference Riccati equations. In this paper, we clarify the essence of Tietze's treatment and make it purely algebraic. As an application, the…
A special class of autonomous algebraic differential equations is studied. No equations in the class have any entire transcendental solutions. In a sense, for almost all equations in the class, transcendental meromorphic solutions can also…