NEW APPLICATIONS OF THE ABEL-LIOUVILLE FORMULA

Abstract

In a recent paper, [1], 2005, the indefinite integrals of a certain type are calculated using some linear homogeneous differential equations of second order with variable coefficients, associated with the integrals. In some simple cases, like the examples considered in this paper, the linear independent solutions of these differential equations are directly calculated relative to elementary functions. In more difficult cases, the power series method must be used. In such situations, it is advisable to use the algebraic symbolic calculus on computer. Examples of this type will be given in a subsequent paper. Because the main formula from which the integrals can be calculated is not rigorous proved in [1], we give here a correct proof based on the Abel-Liouville formula for the differential equations of second order. For completeness, we give here the proof for this formula and some of its applications, necessary to our work. Also, we included two examples.indefinite integrals, second order linear homogeneous differential equations, Abel-Liouville formula