Algorithms of Computer Analysis for Approximate Solutions of Linear ODEs with Polynomial Coefficients

Algorithms in Computer Algebra base on algebraic concepts and aim at finding exact solutions. Computer Analysis gives priority to algorithms using computer algebra systems (CAS) for finding controlled ana-lytic approximate solutions for non closed solvable or only with not acceptable expense solvable problems. In this sense Computer Analysis is the pendant to Numerical Mathematics. The analytic approximate solutions should reflect inherent properties of the given problem and the possible influence of parame-ters. All formula expressions should be simple and transparent, while maintaining an appropriate level of precision. Approximate solutions without adequate error statements are worthless for the practical application. Therefore it is important to control the accuracy by error estimations, which are computed entirely by the computer program, the same way as approximations are determined without requiring user interaction. Efficiency and quality of the algorithms developed in the Computer Analysis depend in a high degree on the possibility to combine the tried methods of numerical computing with symbolic procedures. Many algorithms have their roots in the past and are only now practicable because the fast development of hardware and software technology gives the possibility to implement and improve them. Finding analytic approximate solutions for differential equations is one of the main tasks of the Computer Analysis. In the field of ODEs several algorithms could be developed and tried in the last tw