  Motivated by developing algorithms that decide hypertranscendence of
solutions of extensions of the Bessel differential equation, algorithms
computing the unipotent radical of a parameterized differential Galois group
have been recently developed. Extensions of Bessel's equation, such as the
Lommel equation, can be viewed as homogeneous parameterized linear differential
equations of the third order. In the present paper, we give the first known
algorithm that calculates the differential Galois group of a third order
parameterized linear differential equation.
