Galois Groups of Second and Third Order Linear Differential Equations

Using the representation theory of groups, we are able to give simple necessary and sufficient conditions regarding the structure of the Galois groups of second and third order linear differential equations. These allow us to give simple necessary and sufficient conditions for a second order linear differential equation to have liouvillian solutions and for a third order linear differential equation to have liouvillian solutions or be solvable in terms of second order equations. In many cases these conditions also allow us to determine the group.