On Liouvillian Solutions of Third Order Homogeneous Linear Differential Equations
نویسندگان
چکیده
منابع مشابه
On Second Order Homogeneous Linear Differential Equations with Liouvillian Solutions
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...
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...
متن کاملLiouvillian solutions of linear difference-differential equations
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 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 ...
متن کاملLiouvillian and Algebraic Solutions of Second and Third Order Linear Differential Equations
In this paper we show that the index of a 1-reducible subgroup of the diierential Galois group of an ordinary homogeneous linear diierential equation L(y) = 0 yields the best possible bound for the degree of the minimal polynomial of an algebraic solution of the Riccati equation associated to L(y) = 0. For an irreducible third order equation we show that this degree belongs to f3;6;9;21;36g. Wh...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Journal of Mathematics Research
سال: 2019
ISSN: 1916-9809,1916-9795
DOI: 10.5539/jmr.v11n3p14