This project develops symbolic algorithms to determine the algebraic structure of solutions of differential equations. In particular, the research includes the following: (i) Refine and extend algorithms for computing Galois groups of linear differential equations and algorithms for solving these equations in terms of classes of special functions; (ii) Develop algorithms to decide various properties of the Galois group. For example, give an algorithm to calculate the dimension of the group; (iii) Construct linear differential equations with a given Galois group; (iv) Develop improved algorithms to factor linear equations; and (v) Find first integrals of nonlinear systems of differential equations.
