An algorithm for solving second order linear homogeneous differential equations
Journal of Symbolic Computation
Integration of elementary functions
Journal of Symbolic Computation
Liouvillian Solutions of Linear Differential Equations with Liouvillian Coefficients
Journal of Symbolic Computation
Liouvillian and algebraic solutions of second and third order linear differential equations
Journal of Symbolic Computation
Galois action on solutions of a differential equation
Journal of Symbolic Computation
Journal of Symbolic Computation
On symmetric powers of differential operators
ISSAC '97 Proceedings of the 1997 international symposium on Symbolic and algebraic computation
On second order homogeneous linear differential equations with Liouvillian solutions
Theoretical Computer Science - Special volume on computer algebra
Solving linear ordinary differential equations over C (x, e∫ f(x)dx)
ISSAC '99 Proceedings of the 1999 international symposium on Symbolic and algebraic computation
Galois theory of differential equations, algebraic groups and Lie algebras
Journal of Symbolic Computation - Special issue on differential algebra and differential equations
Solutions of linear ordinary differential equations in terms of special functions
Proceedings of the 2002 international symposium on Symbolic and algebraic computation
Closed form solutions of linear odes having elliptic function coefficients
ISSAC '04 Proceedings of the 2004 international symposium on Symbolic and algebraic computation
Non-liouvillian solutions for second order Linear ODEs
ISSAC '04 Proceedings of the 2004 international symposium on Symbolic and algebraic computation
Notes on triangular sets and triangulation-decomposition algorithms II: differential systems
SNSC'01 Proceedings of the 2nd international conference on Symbolic and numerical scientific computation
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Given a second order linear differential equations with coefficients in a field k=C(x), the Kovacic algorithm finds all Liouvillian solutions, that is, solutions that one can write in terms of exponentials, logarithms, integration symbols, algebraic extensions, and combinations thereof. A theorem of Klein states that, in the most interesting cases of the Kovacic algorithm (i.e when the projective differential Galois group is finite), the differential equation must be a pullback (a change of variable) of a standard hypergeometric equation. This provides a way to represent solutions of the differential equation in a more compact way than the format provided by the Kovacic algorithm. Formulas to make Klein's theorem effective were given in [4, 2, 3]. In this paper we will give a simple algorithm based on such formulas. To make the algorithm more easy to implement for various differential fields k, we will give a variation on the earlier formulas, namely we will base the formulas on invariants of the differential Galois group instead of semi-invariants.