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We give a necessary and sufficient condition for an algebraic ODE to have a rational type general solution. For an autonomous first order ODE, we give an algorithm to compute a rational general solution if it exists. The algorithm is based on the relation between rational solutions of the first order ODE and rational parametrizations of the plane algebraic… (More)

We improve the algorithm given in [3]. Through analysing the structural properties of a first order autonomous ODE with a rational solution, we give a polynomial time algorithm to find a rational general solution if it exists.

For a first order autonomous ODE, we give a polynomial time algorithm to decide whether it has a polynomial general solution and to compute one if it exists. Experiments show that this algorithm is quite effective in solving ODEs with high degrees and a large number of terms.

For a field k with an automorphism σ and a derivation δ, we introduce the notion of liouvillian solutions of linear difference-differential systems {σ(Y) = AY, δ(Y) = BY } over k and characterize the existence of liouvillian solutions in terms of the Galois group of the systems. In the forthcoming paper, we will propose an algorithm for deciding if linear… (More)

In this paper, we give a necessary and sufficient condition for an algebraic ODE to have an algebraic general solution. For a first order autonomous ODE, we give an optimal bound for the degree of its algebraic general solutions and a polynomial-time algorithm to compute an algebraic general solution if it exists. Here an algebraic ODE means that an ODE… (More)

We present a criterion for the existence of telescopers for mixed hypergeometric terms, which is based on multiplicative and additive decompositions. The criterion enables us to determine the termination of Zeilberger's algorithms for mixed hypergeometric inputs.

We present two criteria for the existence of telescopers for bivariate hyperexponential-hypergeometric functions. One is for the existence of telescopers with respect to the continuous variable, the other for telescopers with respect to the discrete one. Our criteria are based on a standard representations of bivariate hyperexponential-hypergeometric… (More)

A normal form is given for integrable linear difference-differential equations {σ(Y) = AY, δ(Y) = BY }, which is irreducible over C(x, t) and solvable in terms of liouvillian solutions. We refine this normal form and devise an algorithm to compute all liouvillian solutions of such kind of systems of prime order.

Most work on finding elementary function solutions for differential equations focussed on linear equations [4, 2, 6, 1, 3]. In this paper, we try to find polynomial solutions to non-linear differential equations. Instead of finding arbitrary polynomial solutions, we will find the polynomial general solutions. For example, the general solution for… (More)

- Ruyong Feng
- ArXiv
- 2013

We present a detailed and simplified version of Hrushovski's algorithm that determines the Galois group of a linear differential equation. There are three major ingredients in this algorithm. The first is to look for a degree bound for proto-Galois groups, which enables one to compute one of them. The second is to determine the identity component of the… (More)