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Many computer algebra algorithms are based on constructing some polynomials and rational functions. Constructing a polynomial often involves finding the bound for its degree and use of the unknown coefficient method. For constructing rational functions cme expands the given functions in partial fractions and from these expansions obtains some hypothesis on the expansion of the function to be found. Thk function itself may be constructed by the uncoefficient method (the numerator of partial fraction plays the role of the unknowns). However the algebraic equation system obtained by the unknown coefficient method may be huge and nonlinear besides. To expand the given rational function one has to factorize the denominator and address the problem of operating with extensions of the constant field. Therefore in designing the algorithms one should “improve upon the pure theoretical considerations. Thk paper is concerned with some way:s for such an improvement with regard to solving the linear ordinary differential equations of the form

Fast algorithms to search for the rational solutions of linear differential equations with polynomial coefficients