Abstract:
 This is an active field of research in computer algebra, namely in the questions of finding algorithms for obtaining closedform (Liouvillian) solutions of linear ODEs with nonconstant coefficients. Many aspects of this field are of big interest for engineering application and may be used for integrability of nonlinear PDEs (the socalled Cintegrability or Darboux integrability). Our exposition of this vast field of research of algorithmic methods for finding closedform solutions of linear ODEs and PDEs will concentrate on the aspects: a) brief description of the algorithms of solution of linear ODEs with variable coefficients (for simplicity the coefficients are assumed to be rational functions) base on the theory of factorization of such linear ODEs. b) classical (Laplace, Moutard, Darboux) methods of integration of linear PDEs with variable coefficients and their recent generalizations. Using a new definition of generalized divisors we obtain analogues of the wellknown theorems of the LoewyOre theory of factorization of linear ordinary differential operators. An application to finding criterions of Darboux integrability of nonlinear PDEs is given. Recently (http://arxiv.org/abs/cs/0609075) a method of obtaining closedform complete solutions of certain secondorder linear partial differential equations with more than two independent variables was given. This method generalizes the classical method of Laplace transformations of secondorder hyperbolic equations in the plane and is based on an idea given by Ulisse Dini in 1902. A number of important applications emerged. For example, in (http://arxiv.org/abs/math/0612793) exact solutions of hyperbolic systems of kinetic equations (master equations), appearing in description of kinetic processes in physics, biology and chemistry were found. Another important application of factorization theory of systems of linear PDEs is solution of defining equations for symmetries and conservation laws of arbitrary nonlinear ODEs and PDEs ("symmetry analysis"). In the joint paper "Factoring Zerodimensional Ideals of Linear Partial Differential Operators" (Ziming Li, Fritz Schwarz, S.P.Tsarev) we presented an algorithm for factoring and finding closedform solutions of linear homogeneous partial differential system with variable coefficients (belong to Q(x,y)), and whose solution space is finitedimensional.
