Integrating factors for second order ODEs

A systematic algorithm for building integrating factors of the form mu(x,y), mu(x,y') or mu(y,y') for second order ODEs is presented. The algorithm can determine the existence and explicit form of the integrating factors themselves without solving any differential equations, except for a linear ODE in one subcase of the mu(x,y) problem. Examples of ODEs not having point symmetries are shown to be solvable using this algorithm. The scheme was implemented in Maple, in the framework of the "ODEtools" package and its ODE-solver. A comparison between this implementation and other computer algebra ODE-solvers in tackling non-linear examples from Kamke's book is shown.Comment: 21 pages - original version submitted Nov/1997. Related Maple programs for finding integrating factors together with the ODEtools package (versions for MapleV R4 and MapleV R5) are available at http://lie.uwaterloo.ca/odetools.ht

Solutions for the General, Confluent and Biconfluent Heun equations and their connection with Abel equations

In a recent paper, the canonical forms of a new multi-parameter class of Abel differential equations, so-called AIR, all of whose members can be mapped into Riccati equations, were shown to be related to the differential equations for the hypergeometric 2F1, 1F1 and 0F1 functions. In this paper, a connection between the AIR canonical forms and the Heun General (GHE), Confluent (CHE) and Biconfluent (BHE) equations is presented. This connection fixes the value of one of the Heun parameters, expresses another one in terms of those remaining, and provides closed form solutions in terms of pFq functions for the resulting GHE, CHE and BHE, respectively depending on four, three and two irreducible parameters. This connection also turns evident what is the relation between the Heun parameters such that the solutions admit Liouvillian form, and suggests a mechanism for relating linear equations with N and N-1 singularities through the canonical forms of a non-linear equation of one order less.Comment: Original version submitted to Journal of Physics A: 16 pages, related to math.GM/0002059 and math-ph/0402040. Revised version according to referee's comments: 23 pages. Sign corrected (June/17) in formula (79). Second revised version (July/25): 25 pages. See also http://lie.uwaterloo.ca/odetools.ht

Incomplete beta-function expansions of the solutions to the confluent Heun equation

Several expansions of the solutions to the confluent Heun equation in terms of incomplete Beta functions are constructed. A new type of expansion involving certain combinations of the incomplete Beta functions as expansion functions is introduced. The necessary and sufficient conditions when the derived expansions are terminated, thus generating closed-form solutions, are discussed. It is shown that termination of a Beta-function series solution always leads to a solution that is necessarily an elementary function

Caracterización fisicoquímica de mieles monoflorales de Euphorbia resinifera

The physicochemical characteristics of Euphorbia resinifera honey were studied. Considering the low water content, the majority of the honeys presented a proper maturity. The values of acidity revealed the absence of inappropriate fermentation, while the low values of hydroxymethylfurfural (0.4–16.48 mg/kg) were suitable for of unprocessed honeys. The average values for electrical conductivity and ashes were 451 µS/cm and 1.6 g/kg, respectively. As for the mineral content, the K was the most abundant element; Ca, Na, Mg, P, S, and Si are all present in differing quantities in the honeys. On the other hand, Principal Components Analysis (PCA) and Stepwise Discriminant Analysis (SDA) were applied to distinguish between three related Euphorbia honey types. PCA showed that the cumulative variance of the two first factors explained approximately 53%. The results of SDA showed that variables with a higher discriminant power were K, C*ab and a*, and 100% of the samples were properly classified in their corresponding class.Los parámetros fisicoquímicos de 29 de mieles monoflorales de Euphorbia resinifera fueron estudiadas. 24 parámetros, incluyendo humedad, pH, acidez (libre, lactónica y total), HMF, cenizas, conductividad eléctrica, monosacáridos (glucosa y fructosa), contenido mineral y parámetros cromáticos fueron analizados. Desde el punto de vista de su calidad las mieles fueron acordes con la legislación Europea en cuanto a contenido en agua, acidez y HMF. Los valores de cenizas y conductividad eléctrica fueron 1,6 g/kg y 451 μS/cm, respectivamente. El contenido en minerales mostró que el K es el elemento más abundante; mientras que Ca, Na, Mg, P, S y Si se presentaron en contenidos intermedios. En cuanto a los valores de los parámetros del color fueron típicos de mieles ámbar claras. Se ha realizado un análisis estadístico multivariante a los datos obtenidos para diferenciar tres especies de mieles de Euphorbia. El análisis discriminante permite diferencia las mieles por su origen botánico siendo el contenido en K, C*ab y la variable cromática a* las variables con mayor poder discriminate, siendo el 100% de las muestras clasificadas correctamente en su grupo