Liouvillian Solutions of Linear Differential Equations with Liouvillian Coefficients
نویسندگان
چکیده
منابع مشابه
Liouvillian Solutions of Linear Differential Equations with Liouvillian Coefficients
Let L(y) = b be a linear differential equation with coefficients in a differential field K. We discuss the problem of deciding if such an equation has a non-zero solution in K and give a decision procedure in case K is an elementary extension of the field of rational functions or is an algebraic extension of a transcendental liouvillian extension of the field of rational functions. We show how ...
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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. We will give an algorithm to decide whether such a system has liouvillian solutions when k = C(x, t), σ(x) = x + 1,...
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We determine all minimal polynomials for second order homogeneous linear diierential equations with algebraic solutions decomposed into in-variants and we show how easily one can recover the known conditions on diierential Galois groups 12,19,25] using invariant theory. Applying these conditions and the diierential invariants of a diierential equation we deduce an alternative method to the algo...
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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. We will give an algorithm to decide whether such a system has liouvillian solutions when k = C(x, t), σ(x) = x+ 1, ...
متن کاملLiouvillian solutions of third order differential equations
The Kovacic algorithm and its improvements give explicit formulae for the Liouvillian solutions of second order linear differential equations. Algorithms for third order differential equations also exist, but the tools they use are more sophisticated and the computations more involved. In this paper we refine parts of the algorithm to find Liouvillian solutions of third order equations. We show...
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ژورنال
عنوان ژورنال: Journal of Symbolic Computation
سال: 1991
ISSN: 0747-7171
DOI: 10.1016/s0747-7171(08)80048-x