Consider a third order linear differential equation L(f) = 0, where <?TeX $L\in \mathbb {Q}(z)[\partial _z]$?> Math 1 . We design an algorithm computing the Liouvillian solutions of L(f) = 0. The reducible cases devolve to the classical case of second order operators, and in the irreducible cases, only finitely many differential Galois groups are possible. The differential Galois group is obtained through optimized computations of invariants and semi-invariants, and if solvable, the solutions are returned as pullbacks and gauge transformations of algebraic generalized hypergeometric function 3F2. The computation time is practical for reasonable size operators.

Let L be a 4th order linear differential operator with coefficients in K(z), with K a computable algebraically closed field. The operator L is called symplectic when up to rational gauge transformation, the fundamental matrix of solutions X satisfies Xt J X=J where J is the standard symplectic matrix. It is called projectively symplectic when it is projectively equivalent to a symplectic operator. We design an algorithm to test if L is projectively symplectic. Furthermore, based on Kovacic's algorithm, we design an algorithm that computes Liouvillian solutions of projectively symplectic operators of order 4. Moreover, using Klein's Theorem, algebraic solutions are given as pullbacks of standard hypergeometric equations.