On second order homogeneous linear differential equations with Liouvillian solutions
We determine all minimal polynomials for second order homogeneous linear differential equations with algebraic solutions decomposed into invariants and we show how easily one can recover the known conditions on differential Galois groups [10, 17, 24] using invariant theory. Applying these conditions and the differential invariants of a differential equation we deduce an alternative method to the algorithms given in [10, 18, 24] for computing Liouvillian solutions. For irreducible second order equations our method determines solutions by formulas in all but three cases.
differential equations liouvillian solutions invariants differential invariants differential galois theory